U.S. patent application number 15/160098 was filed with the patent office on 2016-11-24 for extracting inertial information from nonlinear periodic signals.
The applicant listed for this patent is Lumedyne Technologies Incorporated. Invention is credited to Mark Steven Fralick, John David Jacobs, Charles Harold Tally, IV, Richard Lee Waters.
Application Number | 20160341762 15/160098 |
Document ID | / |
Family ID | 57324393 |
Filed Date | 2016-11-24 |
United States Patent
Application |
20160341762 |
Kind Code |
A1 |
Waters; Richard Lee ; et
al. |
November 24, 2016 |
EXTRACTING INERTIAL INFORMATION FROM NONLINEAR PERIODIC SIGNALS
Abstract
Systems and methods are described herein for extracting inertial
information from nonlinear periodic signals. A system for
determining an inertial parameter can include circuitry configured
for receiving first and second analog signals from first and second
sensors, each sensor responsive to motion of a proof mass. The
system can include circuitry configured for determining a
difference between the first and second analog signals, determining
a plurality of timestamps corresponding to times at which the
difference crosses a threshold, and determining a plurality of time
intervals based on the timestamp. The system can include circuitry
configured for determining a result of applying a trigonometric
function to a quantity, the quantity based on the plurality of time
intervals and determining the inertial parameter based on the
result.
Inventors: |
Waters; Richard Lee; (San
Diego, CA) ; Fralick; Mark Steven; (San Diego,
CA) ; Tally, IV; Charles Harold; (Carlsbad, CA)
; Jacobs; John David; (San Diego, CA) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
Lumedyne Technologies Incorporated |
San Diego |
CA |
US |
|
|
Family ID: |
57324393 |
Appl. No.: |
15/160098 |
Filed: |
May 20, 2016 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
|
|
62164378 |
May 20, 2015 |
|
|
|
Current U.S.
Class: |
1/1 |
Current CPC
Class: |
B81B 2203/051 20130101;
G01P 2015/0865 20130101; G01P 15/125 20130101; G01P 15/13 20130101;
G01C 19/04 20130101; G01D 5/2415 20130101; G01P 15/093 20130101;
G01P 15/097 20130101; B81B 2201/0235 20130101; G01P 21/00 20130101;
G01P 15/0802 20130101; G01P 2015/0814 20130101 |
International
Class: |
G01P 21/00 20060101
G01P021/00; G01P 15/08 20060101 G01P015/08 |
Claims
1. A method for determining an inertial parameter, comprising:
receiving first and second analog signals from first and second
sensors, each sensor responsive to motion of a proof mass;
determining a difference between the first and second analog
signals; determining a plurality of timestamps corresponding to
times at which the difference crosses a threshold; determining a
plurality of time intervals based on the timestamp; determining a
result of applying a trigonometric function to a quantity, the
quantity based on the plurality of time intervals; and determining
the inertial parameter based on the result.
2. The method of claim 1, wherein the first and second sensors are
first and second electrodes, respectively, each interacting with
the proof mass.
3. The method of claim 2, wherein the first and second electrodes
electrostatically interact with the proof mass.
4. The method of claim 1, wherein determining the difference
comprises amplifying the difference between the first and second
analog signals to an analog voltage.
5. The method of claim 1, wherein the quantity comprises a first
quotient of two of the time intervals.
6. The method of claim 5, further comprising: determining a second
result of applying a second trigonometric function to a second
quantity comprising a second quotient of a second two of the time
intervals; and determining a second difference of the result and
the second result.
7. The method of claim 6, further comprising: determining the
multiplicative inverse of the second difference; determining a
product of a pitch of the proof mass and the multiplicative
inverse; and determining an estimate of displacement amplitude of
the proof mass based on the product; wherein the inertial parameter
is the estimate of displacement amplitude.
8. The method of claim 6, further comprising: determining a sum of
the result and the second result; determining a third quotient of
the sum and the difference; determining a fourth quotient of a
first scale factor and a sum of a third two of the time intervals;
determining a square of the fourth quotient; determining a product
of the third quotient, the square, and a second scale factor; and
determining an acceleration of an inertial device comprising the
proof mass; wherein the inertial parameter comprises the
acceleration.
9. The method of claim 8, wherein the second scale factor comprises
a pitch of the proof mass.
10. The method of claim 4, wherein determining the timestamps
comprises: converting the analog voltage to a digital
representation to generate a first periodic digital signal;
interpolating to generate an upsampled digital signal; and
determining timestamps corresponding to times at which the
upsampled digital signal crosses the threshold.
11. The method of claim 4, wherein determining the timestamps
comprises: comparing the analog voltage to a second threshold;
based on determining that the analog voltage has crossed the second
threshold, toggling a rectangular-wave signal between a first value
and a second value; and determining the timestamps based on times
at which the rectangular-wave signal toggles between the first
value and the second value.
12. The method of claim 11, wherein the quantity comprises a phase
shift.
13. The method of claim 1, wherein the quantity comprises a phase
shift.
14. The method of claim 1, wherein each of the plurality of time
intervals is a difference between a respective two of the plurality
of timestamps.
15. A system for determining an inertial parameter, comprising
circuitry configured for: receiving first and second analog signals
from first and second sensors, each sensor responsive to motion of
a proof mass; determining a difference between the first and second
analog signals; determining a plurality of timestamps corresponding
to times at which the difference crosses a threshold; determining a
plurality of time intervals based on the timestamp; determining a
result of applying a trigonometric function to a quantity, the
quantity based on the plurality of time intervals; and determining
the inertial parameter based on the result.
16. The system of claim 15, wherein the first and second sensors
are first and second electrodes, respectively, each interacting
with the proof mass.
17. The system of claim 16, wherein the first and second electrodes
electrostatically interact with the proof mass.
18. The system of claim 15, wherein determining the difference
comprises amplifying the difference between the first and second
analog signals to an analog voltage.
19. The system of claim 15, wherein the quantity comprises a first
quotient of two of the time intervals.
20. The system of claim 19, further comprising circuitry configured
for: determining a second result of applying a second trigonometric
function to a second quantity comprising a second quotient of a
second two of the time intervals; and determining a second
difference of the result and the second result.
21. The system of claim 20, further comprising circuitry configured
for: determining the multiplicative inverse of the second
difference; determining a product of a pitch of the proof mass and
the multiplicative inverse; and determining an estimate of
displacement amplitude of the proof mass based on the product;
wherein the inertial parameter is the estimate of displacement
amplitude.
22. The system of claim 20, further comprising circuitry configured
for: determining a sum of the result and the second result;
determining a third quotient of the sum and the difference;
determining a fourth quotient of a first scale factor and a sum of
a third two of the time intervals; determining a square of the
fourth quotient; determining a product of the third quotient, the
square, and a second scale factor; and determining an acceleration
of an inertial device comprising the proof mass; wherein the
inertial parameter comprises the acceleration.
23. The system of claim 22, wherein the second scale factor
comprises a pitch of the proof mass.
24. The system of claim 18, wherein determining the timestamps
comprises: converting the analog voltage to a digital
representation to generate a first periodic digital signal; and
interpolating to generate an upsampled digital signal; determining
timestamps corresponding to times at which the upsampled digital
signal crosses the threshold.
25. The system of claim 18, wherein determining the timestamps
comprises: comparing the analog voltage to a second threshold;
based on determining that the analog voltage has crossed the second
threshold, toggling a rectangular-wave signal between a first value
and a second value; and determining the timestamps based on times
at which the rectangular-wave signal toggles between the first
value and the second value.
26. The system of claim 25, wherein the quantity comprises a phase
shift.
27. The system of claim 15, wherein the quantity comprises a phase
shift.
28. The system of claim 15, wherein each of the plurality of time
intervals is a difference between a respective two of the plurality
of timestamps.
29. The system of claim 15, further comprising the proof mass and
the first and second sensors.
Description
CROSS REFERENCE TO RELATED APPLICATIONS
[0001] This application claims priority to U.S. Provisional
Application Ser. No. 62/164,378, filed May 20, 2015, the entire
contents of which are hereby incorporated by reference.
BACKGROUND
[0002] Linear inertial sensors, those which use linear signals to
determine inertial information, are subject to error due to drift.
These linear inertial sensors scale linear signals by one or more
predetermined quantities to determine inertial information such as
acceleration or rotation. These predetermined quantities can
account for spring constants, amplifier gain, and other factors.
However, since spring constants, gain, and these other factors can
drift over time, linear inertial sensors can develop an error due
to this drift.
SUMMARY
[0003] Accordingly, systems and methods are described herein for
extracting inertial information from nonlinear periodic
signals.
[0004] A system for determining an inertial parameter can include
circuitry configured for receiving first and second analog signals
from first and second sensors, each sensor responsive to motion of
a proof mass. The system can include circuitry configured for
determining a difference between the first and second analog
signals, determining a plurality of timestamps corresponding to
times at which the difference crosses a threshold, and determining
a plurality of time intervals based on the timestamp. The system
can include circuitry configured for determining a result of
applying a trigonometric function to a quantity, the quantity based
on the plurality of time intervals and determining the inertial
parameter based on the result.
[0005] In some examples, the first and second sensors are first and
second electrodes, respectively, each interacting with the proof
mass.
[0006] In some examples, determining the difference includes
amplifying the difference between the first and second analog
signals to an analog voltage. In some examples, the quantity
includes a first quotient of two of the time intervals.
[0007] In some examples, the system includes circuitry configured
for determining a second result of applying a second trigonometric
function to a second quantity including a second quotient of a
second two of the time intervals and determining a second
difference of the result and the second result.
[0008] In some examples, the system includes circuitry configured
for determining the multiplicative inverse of the second
difference, determining a product of a pitch of the proof mass and
the multiplicative inverse, and determining an estimate of
displacement amplitude of the proof mass based on the product. The
inertial parameter can be the estimate of displacement
amplitude.
[0009] In some examples, the system can include circuitry
configured for determining a sum of the result and the second
result, determining a third quotient of the sum and the difference,
and determining a fourth quotient of a first scale factor and a sum
of a third two of the time intervals. The system can include
circuitry configured for determining a square of the fourth
quotient, determining a product of the third quotient, the square,
and a second scale factor, and determining an acceleration of an
inertial device including the proof mass. The inertial parameter
can include the acceleration. In some examples, the second scale
factor includes a pitch of the proof mass.
[0010] In some examples, determining the timestamps includes
converting the analog voltage to a digital representation to
generate a first periodic digital signal, interpolating to generate
an upsampled digital signal, and determining timestamps
corresponding to times at which the upsampled digital signal
crosses the threshold.
[0011] In some examples, determining the timestamps includes
comparing the analog voltage to a second threshold. Based on
determining that the analog voltage has crossed the second
threshold, circuitry can toggle a rectangular-wave signal between a
first value and a second value. The timestamps can be determined
based on times at which the rectangular-wave signal toggles between
the first value and the second value.
[0012] In some examples, the quantity includes a phase shift. In
some examples, each of the plurality of time intervals is a
difference between a respective two of the plurality of timestamps.
In some examples, the system can further include the proof mass and
the first and second electrodes.
BRIEF DESCRIPTION OF THE DRAWINGS
[0013] The above and other features of the present disclosure,
including its nature and its various advantages, will be more
apparent upon consideration of the following detailed description,
taken in conjunction with the accompanying drawings in which:
[0014] FIG. 1 depicts an inertial device that extracts inertial
information from nonlinear periodic signals, according to an
illustrative implementation;
[0015] FIG. 2 depicts a schematic of an inertial device and
enlarged views that depict fixed and movable teeth of an inertial
device, according to an illustrative implementation;
[0016] FIG. 3 depicts a graph that shows the relationship between
capacitance of a time-domain-switched (TDS) structure and
displacement of a movable element, according to an illustrative
implementation;
[0017] FIG. 4 depicts a graph showing differential capacitance and
displacement of a TDS structure, according to an illustrative
implementation;
[0018] FIG. 5 depicts a block diagram illustrating signal flows of
a system for determining inertial perimeters from an inertial
device, according to an illustrative implementation;
[0019] FIG. 6 depicts a block diagram showing signal flows and
exemplary implementations of the systems and methods described
herein, according to an illustrative implementation;
[0020] FIG. 7 depicts a system using digital control to control
drive velocity, according to an illustrative implementation;
[0021] FIG. 8 depicts a block diagram representing signal flows and
transfer functions of the system depicted in FIG. 7, according to
an illustrative implementation;
[0022] FIG. 9 schematically depicts a feedback loop that represents
the closed-loop feedback of the system depicted in FIG. 7,
according to an illustrative implementation;
[0023] FIG. 10 depicts a Bode plot with a magnitude graph and a
phase graph of the system depicted in FIG. 7, according to an
illustrative implementation;
[0024] FIG. 11 depicts a graph showing the change in oscillation
frequency of the system depicted in FIG. 7 as a function of phase
shift for various quality factors, according to an illustrative
implementation;
[0025] FIG. 12 depicts a graph showing the gain loss of the system
depicted in FIG. 7 as a function of phase shift, according to an
illustrative implementation;
[0026] FIG. 13 depicts a graph illustrating a start-up procedure of
a resonating proof mass, according to an illustrative
implementation;
[0027] FIG. 14 depicts a graph showing sense signals of an inertial
device during resonator start-up, according to an illustrative
implementation;
[0028] FIG. 15 depicts a system that includes two TDS structures
and a graph that depicts capacitance profiles of the TDS
structures, according to an illustrative implementation;
[0029] FIG. 16 depicts a system for determining acceleration with a
transimpedance amplifier (TIA), according to an illustrative
implementation;
[0030] FIG. 17 depicts a graph illustrating signals of the system
depicted in FIG. 16, according to an illustrative
implementation;
[0031] FIG. 18 depicts a system for determining acceleration with a
charge amplifier (CA), according to an illustrative
implementation;
[0032] FIG. 19 depicts a graph illustrating signals of the system
depicted in FIG. 18, according to an illustrative
implementation;
[0033] FIG. 20 depicts a graph illustrating displacement of a proof
mass and TDS timing events, according to an illustrative
implementation;
[0034] FIG. 21 depicts a graph showing various time intervals that
can be extracted from a differential TIA output, according to an
illustrative implementation;
[0035] FIG. 22 depicts a summing block illustrating signal flows
for using an analog-to-digital converter (ADC) to digitally
reproduce an analog input signal, according to an illustrative
implementation;
[0036] FIG. 23 depicts the use of linear interpolation to determine
zero crossings of a digitized signal, according to an illustrative
implementation;
[0037] FIG. 24 depicts a graph illustrating zero-crossings of an
ADC digital output signal, according to an illustrative
implementation;
[0038] FIG. 25 depicts a graph that shows an upsampled AFE output
curve, according to an illustrative implementation;
[0039] FIG. 26 depicts a block diagram illustrating signal flows of
the arccosine algorithm, according to an illustrative
implementation;
[0040] FIG. 27 depicts a block diagram illustrating the signal
flows of the arctangent algorithm, according to an illustrative
implementation;
[0041] FIG. 28 depicts a graph that shows the digital output of the
arctangent algorithm for a low amplitude of proof mass oscillation,
an amplitude that does not result in phase wrap events, according
to an illustrative implementation;
[0042] FIG. 29 depicts a graph showing the output of the arctangent
algorithm when a proof mass has an oscillation amplitude larger
than one-half the pitch distance of a TDS structure, according to
an illustrative implementation;
[0043] FIG. 30 depicts a graph showing a digital output signal of
the arcsine algorithm, where the proof mass has an oscillation
amplitude greater than one-half the pitch, causing phase wraps,
according to an illustrative implementation;
[0044] FIG. 31 depicts a method illustrating phase unwrapping in
the arccosine and arcsine algorithms, according to an illustrative
implementation;
[0045] FIG. 32 depicts an example of phase unwrap error due to
excessive noise at the phase unwrap boundary, according to an
illustrative implementation;
[0046] FIG. 33 depicts capacitive signals of an inertial device
with a proof mass that is driven to amplitudes that do not cause
false phase transitions, according to an illustrative
implementation;
[0047] FIG. 34 depicts capacitance curves of an inertial device
with a proof mass driven at two different amplitudes, according to
an illustrative implementation;
[0048] FIG. 35 illustrates the error reduction from interpolation,
according to an illustrative implementation;
[0049] FIG. 36 depicts an enlarged view of the phase error curves
depicted in FIG. 35, according to an illustrative
implementation;
[0050] FIG. 37 depicts an inertial device with TDS structures that
have four different phase offsets, according to an illustrative
implementation;
[0051] FIG. 38 depicts an in-phase capacitance curve and a
quadrature capacitance curve at a drive amplitude of 2 microns,
according to an illustrative implementation;
[0052] FIG. 39 depicts an in-phase capacitance curve and a
quadrature capacitance curve at a drive amplitude of 7 microns,
according to an illustrative implementation;
[0053] FIG. 40 depicts an in-phase capacitance curve and a
quadrature capacitance curve at a drive amplitude of 12 microns,
according to an illustrative implementation;
[0054] FIG. 41 depicts analog output signals of differential charge
amplifiers at a proof mass oscillation of 7 microns, according to
an illustrative implementation;
[0055] FIG. 42 depicts analog output signals of differential charge
amplifiers at a proof mass oscillation of 4 microns, according to
an illustrative implementation;
[0056] FIG. 43 depicts a ratio of quadrature and in-phase signals,
according to an illustrative implementation;
[0057] FIG. 44 depicts the arctangent of the quadrature to in-phase
signal ratio without unwrapping, according to an illustrative
implementation;
[0058] FIG. 45 depicts a proof mass position after unwrapping,
according to an illustrative implementation; and
[0059] FIG. 46 depicts an enlarged view of a portion of FIG. 45,
showing the difference between a true displacement and a digital
estimate, according to an illustrative implementation.
DETAILED DESCRIPTION
[0060] The systems and methods described herein extract inertial
information from nonlinear periodic signals. In particular, the
systems and methods described herein produce an analog signal that
varies nonlinearly and nonmonotonically in response to monotonic
motion of a proof mass. In some examples, the proof mass is
oscillated periodically, and so the analog signal also oscillates
periodically. Inertial information is extracted from the nonlinear,
nonmonotonic analog signal.
[0061] In some examples, the proof mass is driven to oscillate in a
substantially sinusoidal motion, which causes the analog signal to
oscillate substantially sinusoidally. The proof mass can be driven
in an open-loop manner, or it can be driven by an analog or a
digital closed-loop drive. One way to produce a nonmonotonic signal
from a monotonic motion of the proof mass is to oscillate a surface
of the proof mass relative to an opposing surface, both surfaces
having some nonplanarity. In some examples, the opposing surface is
located on a frame of the inertial device, such that the opposing
surface experiences the same acceleration as the inertial
device.
[0062] One example of a surface nonplanarity is a single asperity,
or a tooth. Teeth on opposing surfaces can be aligned when the
proof mass is in a rest position, or the teeth can be shifted with
respect to each other at rest. As the proof mass moves with respect
to the opposing surface in a motion that maintains the nominal gap
between the proof mass and the opposing surface, the spacing
between the tips of the teeth changes. As the teeth approach and
then move past each other, the spacing between the respective teeth
varies nonmonotonically, because it decreases and then increases.
The spacing changes nonmonotonically even though the motion of the
proof mass is monotonic over this region. This nonmonotonic change
in spacing between the teeth produces an analog signal that also
changes nonmonotonically based on a monotonic motion of the proof
mass. The analog signal can be received by a sensor that responds
to motion of the proof mass. The sensor can comprise an electrode.
The electrode can electrostatically interact with the proof mass.
The analog signal can be produced as a result of electrostatic
interaction between the proof mass and the opposing surface.
Depending on the configuration of the sensor, the analog signal can
be a capacitance, a capacitive current, an inductance, an inductive
current, a tunneling current, an optical signal, an electromagnetic
signal, or another similar signal. An electrical voltage can be
applied between the proof mass and the opposing surface to aid in
generating the analog signal.
[0063] There are other possible ways to create a spatial frequency
that is higher than the drive frequency and that would be to use
coupled oscillator systems where sums and differences of the two
resonator frequencies are generated. For the coupled oscillator
example, the geometric dimension is tied to the length, width and
thickness of the complaint spring structures used to establish the
resonant frequencies of each of the coupled oscillators.
[0064] Another possibility is an optical shuttering system (where
optics are used instead of electrostatics). The shuttering
mechanism is attached to the oscillating proof mass and a sensor is
positioned to detect light from a source. The shuttering mechanism
modulates the intensity of the transmitted light. Changes in the
light transmission resulting from movement of the shuttering
mechanism with the proof mass are sensed by the optical sensor. In
this case, as a result of changes in the position of the shutter,
there can be an increase, such as a doubling, in modulation
frequency as the light is passed through the shutter relative to an
oscillation of the proof mass, such as two times per oscillation
cycle of the proof mass. The sensor responds to the changes in
transmission resulting from motion of the proof mass, such that the
reference mass interacts with the sensor, and produces a resulting
analog signal.
[0065] Alternatively, again using optics, is to create an optically
resonant cavity such as a Fabry-Perot wherein one of the mirrors is
attached to the proof mass. If the proof mass oscillates such that
the cavity spacing between the mirrors changes, and if the
oscillation amplitude is large enough, the cavity will spatially
pass multiples of n*.lamda./2 where .lamda., is the wavelength of
light and n is the index of refraction of the optical cavity. Every
time the spatial mirror gap reaches n*.lamda./2, a maximum in
optical transmission occurs. So as long as the drive amplitude
>n*.lamda./2 multiple max or min values will be reached every
oscillation cycle. In this way, the optical sensor will sense the
variation in transmission resulting from motion of the proof mass,
such that the reference mass interacts with the sensor, and the
sensor responds to the position (and motion) of the proof mass to
produce an analog signal. For the optical resonator, the
geometrical dimension is tied to the wavelength of light used.
[0066] In some examples, it is desirable to amplify the analog
signal by using a proof mass with an array of teeth and an opposing
surface with another array of teeth. Each array of teeth is
regularly spaced, with a pitch defining the distance between
adjacent teeth in the array. The two arrays of teeth have the same
pitch so that amplification of the produced signal is maximized. In
other words, there exists a relative position of the proof mass
such that all of the teeth in the array on the proof mass are at
the minimum separation from the opposing teeth in the array on the
opposing surface. In some examples, the produced signal can be
amplified further by interdigitating the proof mass with the
opposing surface and arranging arrays of teeth on each of the
interdigitated surfaces of the proof mass and the opposing
surface.
[0067] The teeth can be rectangular, triangular, or another shape.
The shape of the teeth determines the specific relationship between
the produced signal and the motion of the proof mass, but does not
change the nonmonotonicity.
[0068] An analog front end (AFE) converts the analog signal
produced by the teeth to an analog voltage signal. The AFE does
this by generating an analog voltage that is linearly proportional
to the analog signal produced by the teeth. Thus, the analog
voltage signal is also nonlinear and nonmonotonic. The AFE can be
selected based on the type of analog signal to be measured. If the
analog signal to be measured is a capacitance, the AFE can be a
capacitance-to-voltage (C-to-V) converter such as a charge
amplifier (CA) or a bridge with a general impedance converter
(GIC). If the produced signal is a current such as a capacitive
current or a tunneling current, the AFE can include a current
amplifier such as a transimpedance amplifier (TIA). If the analog
signal to be measured is optical, the AFE can include an optical
device such as a photodiode or a charge coupled device. If the
produced signal is electromagnetic, the AFE can include an
antenna.
[0069] In some examples the inertial devices includes a
time-to-digital converter (TDC) to convert the analog voltage
signal to a digital signal. The TDC measures times at which the
analog signal crosses certain thresholds, such as when the analog
signal experiences maxima, minima, zeros, or other values. In some
examples, the TDC produces a binary output that switches between
two values when the analog voltage signal crosses these
thresholds.
[0070] In some examples, the inertial device uses an
analog-to-digital converter (ADC) to convert the analog voltage
signal to a digital signal. The digital signal can then be used to
determine inertial information. In some examples, the inertial
device can include digital circuitry which extracts inertial
information from the digital signal produced by the ADC or the
TDC.
[0071] FIG. 1 depicts an inertial device 100 that extracts inertial
information from nonlinear periodic signals. The inertial device
100 includes a proof mass 102 that is connected to anchors 112a,
112b, 112c, and 112d (collectively, anchors 112) by springs 110a,
110b, 110c, 110d (collectively, springs 110), respectively. FIG. 1
also depicts a coordinate system 122 with an x axis, a y axis
perpendicular to the x axis, and a z axis perpendicular to each of
the x and y axes. The proof mass 102 is driven along the x axis by
drive combs 114a and 114b (collectively, drive combs 114). An AC
voltage applied to the drive combs 114 causes the proof mass 102 to
oscillate along the x axis. The inertial device 100 includes sense
combs 118a, 118b, 118c, and 118d (collectively sense combs 118)
used to detect motion of the proof mass 102. As the proof mass 102
oscillates along the x axis, the capacitance between the sense
combs 118 and the proof mass 102 varies. This varying capacitance
causes a capacitive current to flow when a DC sense voltage is
applied between the sense combs 118 and the proof mass 102. This
capacitive current, which is proportional to the position of the
proof mass 102, can be used to determine drive amplitude and
velocity of the proof mass 102. The sense combs 118 are linear in
that they produce an analog output signal (e.g., current or
capacitance) that is a monotonic and linear (or substantially
linear) function of the position of the proof mass 102.
[0072] The inertial device 100 can include a digital closed-loop
drive which regulates the amplitude of the motion of the proof mass
102 to a desired value. The digital closed-loop drive can use the
drive amplitude and velocity of the proof mass determined using the
sense combs 118. The digital (closed-loop drive) compares the
measured motion of the proof mass 102 to the desired value and
regulates the voltage applied to the drive combs 114 to maintain
the amplitude of the proof mass 102 at the desired value.
[0073] The proof mass 102 includes arrays of movable teeth 104a and
104b (collectively, movable teeth 104). The movable teeth 104 are
spaced along the x axis. The inertial device 100 includes fixed
beams 108a, 108b, 108c, and 108d (collectively, fixed beams 108).
The fixed beams 108 include arrays of fixed teeth 106a, 106b, 106c,
and 106d (collectively, fixed teeth 106), respectively. The fixed
teeth 106 are spaced along the x axis and adjacent to the movable
teeth 104. The fixed teeth 106 and the movable teeth 104
electrostatically interact with each other. As teeth of the movable
teeth 104 align with adjacent teeth of the fixed teeth 106,
capacitance between the beams 106 and 108 is at a maximum. As teeth
of the movable teeth 104 align with gaps between teeth of the fixed
teeth 106, capacitance between the beams 106 and 108 is at a
minimum. Thus, as the proof mass 102 moves monotonically along the
x axis, capacitance between the proof mass 102 and the fixed beams
108 varies nonmonotonically, increasing as teeth align with
adjacent teeth and decreasing as teeth align with gaps. In some
examples, as is depicted in FIG. 1, the inertial device includes
arrays of teeth arranged with phase offsets. In the example
depicted in FIG. 1, when the proof mass 102 is in its neutral
position, teeth of the movable teeth 104 align with teeth of the
fixed teeth 106b and 106c. When the proof mass 102 is in the
neutral position, as shown in FIG. 1, teeth of the fixed teeth 106a
and 106d do not align with teeth of the movable teeth 104, but
instead align with the centers of gaps between respective teeth of
the movable teeth 104. In this configuration, when the teeth 106b
and 106c experience a maximum in capacitance, the teeth 106a and
106d experience a minimum in capacitance, and vice-versa. Likewise,
as the capacitances of the teeth 106b and 106c are increasing, the
capacitances of the teeth 106a and 106d are decreasing, and
vice-versa. These differences in phase between arrays of teeth can
be used as described herein to perform a differential measurement
of capacitance between the proof mass 102 and the fixed beams
108.
[0074] The inertial device 100 includes a device layer comprising
the features depicted in FIG. 1. The inertial device 100 also
includes a top layer (not shown) above the device layer and a
bottom layer (not shown) below the device layer. The anchors 112,
the fixed beams 108, the drive combs 114, and the sense combs 118
are connected to one or both of the top layer (not shown) and the
bottom layer (not shown). The proof mass 102 can move freely within
the plane of the device layer.
[0075] In some examples, the proof mass 102 is at a ground voltage,
as it is electrically connected to the anchors 112 by the springs
110. In these examples, the anchors 112 are grounded through their
connection to the bottom layer (not shown) or the top layer (not
shown). In some examples, a DC voltage is applied to the fixed
beams 108. In some examples, the DC voltage applied is 2.5 V. In
some examples, DC voltages of opposite polarities are applied to
the sense combs 118 to enable a differential capacitance
measurement. In some examples, a voltage of +2.5 V is applied to
the sense combs 118c and 118d, and a DC voltage of -2.5 V is
applied to the sense combs 118a and 118b. In some examples, the AC
voltages applied to the respective drive combs 114a and 114b are of
equal amplitudes, but 180.degree. out of phase. In these examples,
the drive combs 114 alternately electrostatically attract, or
"pull," the proof mass.
[0076] FIG. 2 depicts a schematic of an inertial device 202 and
enlarged views 206, 210, and 214 that depict fixed and movable
teeth of the inertial device 202. The view of the inertial device
202 schematically depicts a proof mass and movable and fixed teeth.
The view of inertial device 202 includes an area of interest 204
that includes both movable teeth connected to the proof mass and
fixed teeth connected to an anchor. The enlarged view 206 is an
enlarged view of features of the area of interest 204. The enlarged
view 206 depicts a time-domain-switched (TDS) structure 207 that
includes a fixed element 216 and a movable element 220. The movable
element 220 is connected to the proof mass 203, while the fixed
element 216 is anchored to a bottom layer (not shown) and/or a top
layer (not shown) of the inertial device 202. The fixed element 216
includes a plurality of fixed beams, including a fixed beam 218.
Likewise, the movable element 220 includes a plurality of movable
beams, including a movable beam 222. The enlarged view 206 also
includes an area of interest 208.
[0077] The enlarged view 210 is an enlarged view of the area of
interest 208 and depicts fixed and movable beams, including the
fixed beam 218 and the movable beam 222. The enlarged view 210 also
includes an area of interest 212.
[0078] The enlarged view 214 is an enlarged view of the area of
interest 212. The enlarged view 214 depicts the fixed beam 218 and
the movable beam 222. The fixed beam 218 includes fixed teeth 226a
and 226b (collectively, fixed teeth 226). The movable beam 222
includes movable teeth 224a and 224b (collectively, movable teeth
224). The centers of the fixed teeth 226 are separated by a pitch
distance 228, and the centers of the movable teeth 224 are
separated by the same pitch distance 228. Furthermore, the teeth of
the movable beam 222 and the teeth of the fixed beams 218 have the
same widths and have the same gaps between adjacent teeth.
[0079] FIG. 3 depicts a graph 300 that shows the relationship
between capacitance of the TDS structure 207 (FIG. 2) and
displacement of the movable element 220 (FIG. 2). The graph 300
includes a capacitance curve 302 representing capacitance as a
function of displacement of the movable element 220 (FIG. 2). The
capacitance curve 302 is centered at a middle level 306 and ranges
between a maximum level 304 and a minimum level 308. The
capacitance curve 302 reaches the maximum level 304 when the teeth
of the movable element 220 (FIG. 2) are aligned with teeth of the
fixed element 216 (FIG. 2). The capacitance curve 302 reaches the
minimum level 308 when the teeth of the movable element 220 (FIG.
2) are anti-aligned (e.g., aligned with gaps between) the teeth of
the fixed element 216 (FIG. 2). While changes in voltage applied
between the fixed element 216 (FIG. 2) and the movable element 220
(FIG. 2) can affect the amplitude of the capacitance curve 302, the
capacitive curve 302 and the maximum level 304 and the minimum
level 308, changes in voltage will not affect the displacement at
which the capacitive curve reaches a maximum or minimum. The
capacitance curve 302 reaches maximum and minimum levels at
permanently fixed positions that are defined during fabrication of
the inertial device 202 (FIG. 2), because the capacitance curve 302
reaches these level when teeth are aligned or anti-aligned. Thus,
the displacement at which the capacitive curve reaches the maximum
level 304 or the minimum level 308 are unaffected by drift in the
voltage applied between the fixed element 216 (FIG. 2) and the
movable element 220 (FIG. 2).
[0080] The graph 300 also depicts displacement levels that include
a -P displacement level 318, a -3P/4 displacement level 326, a -P/2
displacement level 314, a -P/4 displacement level 322, a 0
displacement level 310, a +P/4 displacement level 320, a +P/2
displacement level 312, a +3P/4 displacement level 324, and a +P
displacement level 316. The graph 302 reaches the maximum
capacitance level 304 at the displacement levels 318, 310, and 316,
and reaches the minimum capacitance level 308 at the displacement
levels 314 and 312. The capacitance curve 302 intersects the middle
level 306 at the displacement levels 326, 322, 320, and 324. Thus,
the capacitance curve 302 experiences maxima when the movable
element 220 (FIG. 2) has moved integer multiples of the pitch
distance from its neutral position. The capacitance curve 302
experiences minima when the movable element 220 (FIG. 2) has moved
one-half the pitch distance from its neutral position in either
direction.
[0081] The movable element 220 (FIG. 2) is resonated with the
respect to the fixed element 206 (FIG. 2). As the movable element
oscillates in sinusoidal motion, the capacitance varies
periodically according to the capacitance curve 302. The periodic
variation depends on factors including the shape of the teeth of
the movable element 220 and the fixed element 206, the size of gaps
between the teeth, and manufacturing variations. In some examples,
the periodic variation is sinusoidal, in some examples, the
periodic variation is semi-sinusoidal, and in some examples, the
periodic variation is not sinusoidal. Because the movable element
220 (FIG. 2) is connected to the proof mass 203 (FIG. 2), an
acceleration applied to the inertial device will affect the motion
of the proof mass 203 (FIG. 2) and the movable element 220 (FIG.
2). The acceleration applied to the inertial device 202 (FIG. 2)
will affect the capacitance curve 302 by shifting the maximum level
304, the middle level 306, and the minimum level 308 in proportion
to the magnitude of the acceleration. This offset in the
capacitance curve 302 can be measured and used to determine the
applied acceleration. The applied acceleration is measured with
respect to the pitch distance, which is a fixed spatial constant
defined by the fabrication process.
[0082] FIG. 4 depicts a graph 400 showing differential capacitance
and displacement of a TDS structure such as the TDS structure 207
(FIG. 2). The graph 400 includes a differential TDS capacitance
curve 404 and a displacement curve 402. The displacement curve 402
represents the motion of a proof mass (e.g., 102 (FIG. 1)) as a
function of time. The differential TDS capacitance curve 404
represents a difference in capacitance between an in-phase TDS
structure (e.g., 105b, 105c (FIG. 1)) and an out-of-phase TDS
structure (e.g., 105a, 105d (FIG. 1)). The graph 400 includes times
406, 412, and 418 at which the differential TDS capacitance curve
404 reaches a maximum and at which a proof mass (e.g., 102 (FIG.
1)) is displaced at integer multiples of the pitch distance from a
neutral position. The graph 400 also includes times 408 and 416 at
which the proof mass is displaced by one-half the pitch distance
from its neutral position and at which the differential TDS
capacitance curve 404 reaches a minimum. The graph 400 also
includes times 410, 414, 424, and 426 at which the proof mass is
displaced by one-half the pitch distance from its neutral position
and at which the differential TDS capacitance curve crosses a zero
level. The graph 400 also includes a displacement level 420 at
which the displacement curve 402 reaches a minimum and a
displacement level 422 at which the displacement curve 402 reaches
a maximum. The differential TDS capacitance curve 404 also reaches
a maximum at the times 420 and 422, but the differential TDS
capacitance at the times 420 and 422 is lower than at the times
406, 412, and 418. The differential TDS capacitance curve 404 only
reaches a maximum at the times 420 and 422 because the proof mass
(e.g., 102 (FIG. 1)) reverses direction at these times. Thus, the
maximum in capacitance at the times 420 and 422 is not defined by
the pitch of the TDS structures, but instead by the drive amplitude
of the proof mass (e.g., 102, 203 (FIGS. 1 and 2)). As, as such,
the maximum in capacitance at the times 420 and 422 is not used for
determining acceleration. Equation 1 illustrates the dependence on
time and displacement of the differential TDS capacitance curve
404.
C(t)=C[x(t)] [1]
[0083] Equation 2 shows the relationship between displacement and
time of the displacement curve 402.
x(t)=Asin(.omega..sub.0(t)+ . . . x(t).sub.INERTIAL [2]
[0084] As shown in equation 2, the displacement curve 402 is
affected by a sinusoidal drive component and an inertial
component.
[0085] The time interval T.sub.1 428 corresponds to the interval
between times 410 and 426 of successive crossings of the -P/4
level. The time interval T.sub.2 430 corresponds to the interval
between times 414 and 424 of successive crossings of the +P/4
level. The time intervals T.sub.1 428 and T.sub.2 430 can be used
as shown in equations 3-6 to determine oscillation offset .DELTA.
of the proof mass (e.g., 102 (FIG. 1)) and acceleration A of the
inertial device (e.g., 100 (FIG. 1)).
.DELTA. - A cos ( w 0 T 1 2 ) = + P 4 [ 3 ] .DELTA. - A cos ( w 0 T
2 2 ) = - P 4 [ 4 ] A = P / 2 [ cos ( .omega. 0 T 1 2 ) - cos (
.omega. 0 T 2 2 ) ] [ 5 ] .DELTA. = A 2 [ cos ( w 0 T 1 2 ) + cos (
w 0 T 2 2 ) ] [ 6 ] ##EQU00001##
[0086] FIG. 5 depicts a block diagram 500 illustrating signal flows
of a system for determining inertial perimeters from an inertial
device (e.g., 100, 202 (FIGS. 1 and 2)). The block diagram 500
includes a drive controller 502 that applies a drive voltage 504 to
a MEMS TDS sensor 506. The MEMS TDS sensor 506 can be the inertial
device 100 (FIG. 1) or the inertial device 202 (FIG. 2), and the
drive voltage 504 can be applied to the drive combs 114 (FIG. 1) to
oscillate a proof mass (e.g., 102). A drive sense capacitance 508
of the MEMS TDS sensor 506 is detected by a drive sense pickoff AFE
510. The drive sense capacitance 508 can be a capacitance of the
sense combs 118 (FIG. 1). The AFE 510 generates an analog output
512 that is proportional to an estimate of drive amplitude. The
analog output 512 is received by the drive controller 502 and used
to adjust the drive voltage 504 to regulate the motion of the proof
mass (e.g., 102, 203 (FIGS. 1 and 2)) to a constant amplitude. The
MEMS TDS sensor 506 generates a TDS capacitance 514 that is
detected by a TDS AFE 516. The TDS AFE 516 can be a charge
amplifier (CA), a transimpedence amplifier (TIA), or a bridge with
a general impedance converter (GIC). The TDS AFE 516 produces an
analog output 518 that is a representation of the TDS capacitance
signal 514. The analog output 518 is digitized by digitization
circuitry 520. The digitization circuitry 520 can be a
time-to-digital converter (TDC) or an analog-to-digital converter
(ADC). The digitization circuitry 520 produces a digital signal 522
that is a digital representation of the TDS capacitance signal 514.
The digital signal 522 is received by digital circuitry 524 that
implements one or more TDS inertial algorithms (including the
cosine, arcsine, arccosine, and arctangent algorithms) to determine
amplitude, frequency, and acceleration information 526 of the MEMS
TDS sensor 506.
[0087] FIG. 6 depicts a block diagram 600 showing signal flows and
exemplary implementations of the systems and methods described
herein. The block diagram 600 schematically depicts a MEMS
structure 602 that includes a proof mass 608 and TDS structures 604
and 606. The TDS structure 604 is an in-phase TDS structure and the
TDS structure 606 is an out-of-phase TDS structure. In some
examples, the out-of-phase TDS structure 606 may be shifted by a
portion of the pitch distance. The proof mass 608 oscillates with
its motion schematically depicted by graph 610. As a result of the
motion of the proof mass 608, the TDS structures 604 and 606
produce non-linear capacitive signals 611 and 613, respectively.
The non-linear capacitive signals 611 and 613 are illustrated by
graphs 612 and 614, respectively. The non-linear capacitive signals
611 and 613 are received by a differential AFE 616. The
differential AFE 616 can be CA 618 or a TIA 620. The differential
AFE 616 outputs an analog signal 626 that corresponds to the
difference between the non-linear capacitive signals 611 and 613.
If the AFE 616 is a CA 618, the analog signal 626 is a voltage that
represents capacitance of the TDS structures 604 and 606, and is
illustrated by the graph 622. If the AFE 616 is a TIA 620, the
analog signal 626 is a voltage that represents a time rate of
change of capacitance of the TDS structures 604 and 606, and is
illustrated by graph 624.
[0088] In some examples, the analog signal 626 is received by a
comparator 628 that outputs an rectangular-wave signal 629 based on
comparing the analog signal 626 to one or more thresholds. If the
AFE 616 is a CA 618, the rectangular-wave signal 629 represents
times at which the capacitance of the TDS structure 604 and 606
crosses the one or more thresholds and is illustrated by graph 630.
If the AFE 616 is a TIA 620, the rectangular-wave signal 629
represents times at which the time rate of change of capacitance of
the TDS structures 604 and 606 crosses the one or more thresholds
and is illustrated by graph 634. The rectangular-wave signal 629 is
received by a time-to-digital converter (TDC) which provides
digital signals representing timestamps of threshold crossings to
digital circuitry that implements a cosine algorithm to determine
acceleration of the inertial device 602.
[0089] In some examples, the analog signal 626 is received by an
ADC 634. The ADC 634 generates a digital signal 635 that represents
the analog signal 626. If the AFE 616 is a CA 618, the digital
signal 635 represents a capacitance of the TDS structures 604 and
636. If the AFE 616 is a TIA 620, the digital signal 635 represents
a time rate of change of capacitance of the TDS structures 604 and
606, and is illustrated by graph 638. In some examples, digital
circuitry 640 receives the digital signal 635 and performs digital
interpolation to determine times at which the digital signal 635
crosses a threshold, and then implements the cosine algorithm to
determine proof mass displacement and/or acceleration of the
inertial device 602 based on the timestamps. In some examples,
digital circuitry 642 receives the digital signal 635 and
implements an arctangent algorithm to determine proof mass
displacement and/or acceleration of the inertial device 602 based
on the digital signal 635. In some examples, digital circuitry 644
receives the digital signal 635 and implements an arccosine or an
arcsine algorithm to determine proof mass displacement and/or
acceleration of the inertial device based on the digital signal
635. Accordingly, a CA 618 or a TIA 620 can be used in conjunction
with a comparator 628 or an ADC 634 and digital circuitry to
implement the cosine algorithm, the arctangent algorithm, the
arccosine algorithm, or the arcsine algorithm.
[0090] FIG. 7 depicts a system 700 using digital control to control
drive velocity. The system 700 includes an oscillating structure
715. The oscillating structure 715 can be the drive frame 120 (FIG.
1). The oscillating structure 715 includes drive capacitors 718a
and 718b (collectively, drive capacitors 718) that cause the
oscillating structure 715 to oscillate. The oscillating structure
715 also includes sense capacitors 717a and 717b (collectively,
sense capacitors 717). The sense capacitors 717 produce current
signals 720 and 722 which are provided to a transimpedance
amplifier system 712. The transimpedance amplifier system produces
differential output signals that are provided to a fixed gain
amplifier 728 and a low-pass filter 713. The output of the low-pass
filter 713 is provided to a comparator 714 which produces a
rectangular-wave drive sync signal.
[0091] The outputs of the fixed gain amplifier 728 are provided to
a kick-start subsystem 740. The kick-start subsystem 740 includes a
set of switches and a high voltage kick-start pulse sequence used
to initiate oscillations of the oscillating structure 715. When the
oscillating structure 715 is oscillating in steady state, the
kick-start subsystem 740 simply passes the outputs of the fixed
gain amplifier 728 on as the drive signals 724 and 726. The drive
signals 724 and 726 are provided to the drive capacitors 718 and
cause the drive capacitors 718 to drive the oscillating structure
715 into oscillation.
[0092] The system 700 includes a digital automatic gain control
loop 730. The digital automatic gain control loop 730 includes
amplitude computation circuitry 732. The amplitude computation
circuitry 732 uses time intervals from nonlinear periodic
capacitors such as the TDS structures (e.g., 105, 207, 506, 604,
606, 1501, 1503, 1606, 1608, 1806, 1808 (FIGS. 1, 2, 5, 6, 15, 16,
and 18)) to determine amplitude of the oscillations of the
oscillating structure 715. The amplitude computation circuitry 732
produces an amplitude output which is subtracted from a desired
amplitude at block 734 to produce an error signal which is provided
to a digital controller 736. The digital controller 736 can use
proportional-integral-derivative (PID) control to adjust a bias
voltage 738 that is provided to the common mode offset terminals of
the amplifier 728 and the amplifier in the amplifier system 712. By
using digital control to adjust the output common mode voltage
level of the amplifiers, the system 700 maintains a desired drive
amplitude. The system 700 could also maintain a desired drive
frequency. By controlling drive amplitude and/or frequency, the
velocity of the oscillating structure 715 is controlled.
[0093] FIG. 8 depicts a block diagram 800 representing signal flows
and transfer functions of the system 700 (FIG. 7). The block
diagram 800 includes a MEMS dynamics block 802 reflecting the
transfer function of force into oscillator velocity. The oscillator
velocity from the MEMS dynamics block 802 is provided to a sense
capacitor block 804 which includes a transfer function for
transferring oscillator velocity into sense current. The sense
current produced by the sense capacitor block 804 is provided to a
transimpedance amplifier block 806 which converts sense current
into a sense voltage. The sense voltage is provided to a fixed gain
amplifier block 808 which converts the sense voltage into an AC
voltage to power the drive capacitors. The AC voltage is provided
to a symmetric drive block 810 which represents the drive
capacitors. The symmetric drive 810 transforms the AC voltage into
a force acting on the oscillator. The block diagram 800 also
includes TDC timing values 812 provided to an amplitude computation
block 814. The amplitude computation block 814 computes an
amplitude which is provided to a summing block 816. The summing
block 816 subtracts the computed amplitude from an amplitude
setpoint and provides an output error to a PID controller block
818. The PID controller block 818 provides a drive voltage control
setting to the sense capacitor block 804 and the symmetric drive
block 810.
[0094] FIG. 9 schematically depicts a feedback loop 900 that
represents the closed-loop feedback of the system 700 (FIG. 7). The
feedback loop 900 includes a drive voltage block 910 that provides
a voltage to drive capacitors. The drive voltage produced by the
drive voltage block 910 results in a force 902 that is in phase
with the drive voltage 910. The force 902 produces a proof mass
displacement 904 that has a -90.degree. phase offset from the force
902. The proof mass displacement 904 produced a sense current 906
that has a +90.degree. phase offset from the proof mass
displacement 904. Thus, the sense current 906 is in phase with the
drive voltage 910 and the force 902. A transimpedance amplifier 908
produces a voltage based on the sense current 906 that is
approximately in phase with the sense current 906. The voltage
produced by the transimpedance amplifier 908 is provided to the
drive voltage block 910, which adjusts the drive voltage
accordingly. Thus, appropriate phase offsets are maintained
throughout the feedback loop 900.
[0095] Interdigitated electrode (IDE) capacitors provide a means
for driving and sensing inertial motion of a MEMS proof mass. The
drive combs 114 (FIG. 1) and the sense combs 118 (FIG. 1) are
examples of IDE capacitors. Expressions for the IDE capacitance and
associated gradient terms of each of the drive combs 114 (FIG. 1)
and each of the sense combs 118 (FIG. 1) are shown in equations
7-14.
[0096] The capacitance and gradient in capacitance of the left-side
drive combs (e.g., 114a (FIG. 1)) is given by equations 7 and
8.
C DL = N h ( g o - x ) d + C fringeDL [ F ] [ 7 ] .gradient. C DL =
C DL x = - N h d = - C o g o .ident. - .gradient. C D [ F m ] [ 8 ]
##EQU00002##
[0097] The capacitance and gradient in capacitance of the
right-side drive combs (e.g., 114b (FIG. 1)) is given by equations
9 and 10.
C DR = N h ( g o + x ) d + C fringeDR [ F ] [ 9 ] .gradient. C DR =
C DR x = + N h d = + C o g o [ F m ] [ 10 ] ##EQU00003##
[0098] The capacitance and gradient in capacitance of the left-side
sense combs (e.g., 118a (FIG. 1)) is given by equations 11 and
12.
C SL = N h ( g o - x ) d + C fringeSL [ F ] [ 11 ] .gradient. C SL
= C SL x = - N h d = - C o g o .ident. - .gradient. C S [ F m ] [
12 ] ##EQU00004##
[0099] The capacitance and gradient in capacitance of the
right-side sense combs (e.g., 118b (FIG. 1)) is given by equations
13 and 14.
C SR = N h ( g o - x ) d + C fringeSR [ F ] [ 13 ] .gradient. C SR
= C SR x = + N h d + C o g o .ident. - .gradient. C D [ F m ] [ 14
] ##EQU00005##
[0100] It is assumed that fringe capacitance does not vary with
position and that nonlinear capacitance contributions (e.g.,
parallel plate effects) are negligible. The differential force
applied to the proof mass (e.g., 102, 203, 608 (FIGS. 1, 2, and 6))
subject to a symmetric sinusoidal drive voltage with static DC
offset is given by equations 15-19.
v AC = V AC cos ( .omega. D t ) ) [ 15 ] Force = - W Total x [ N ]
[ 16 ] Force = 1 2 .gradient. C DL ( V DC - V Proof - v AC ) 2 + 1
2 .gradient. C DR ( V DC - V Proof + v AC ) 2 + 1 2 .gradient. C SL
( V DC - V Proof ) 2 + 1 2 .gradient. C SR ( V DC - V Proof ) 2 [
17 ] Force = - 1 2 .gradient. C D ( ( V DC - V Proof ) 2 + v AC 2 -
2 ( V DC - V Proof ) v AC ) + 1 2 .gradient. C D ( ( V DC - V Proof
) 2 + v AC 2 + 2 ( V DC - V Proof ) v AC ) [ 18 ] Force = 2
.gradient. C D ( V DC - V Proof ) v AC = 2 C o g o ( V DC - V Proof
) v AC [ 19 ] ##EQU00006##
[0101] This implicitly assumes that the forces produced by the
drive sense and TDS pick-off capacitor pairs approximately cancel
resulting in negligible influence on the overall forcing vector.
The collective force is in-phase with the applied AC drive voltage
excitation.
[0102] Applied force leads to a mechanical displacement. Equation
20 below generally provides an adequate representation of the
mechanical behavior of the proof mass.
H ( s ) = .DELTA. x Force = L { .DELTA. x m x } = 1 / m s 2 + s
.omega. o Q + .omega. o 2 [ 20 ] ##EQU00007##
[0103] At resonance, the transfer of force to displacement imparts
a -90.degree. phase shift, as shown in equation 21.
H ( j.omega. o ) = - j Q m .omega. o 2 [ 21 ] ##EQU00008##
[0104] Motion of the proof mass causes a change in capacitance of
the drive sense electrodes (e.g., 118 (FIG. 1)). The drive sense
electronics maintain a fixed bias voltage across the drive sense
electrodes, which in turn produces a detectable motion-induced
current as shown in equations 22 and 23.
i.sub.sense={dot over (q)}.sub.sense=
.sub.S(V.sub.DC-V.sub.Proof)+C.sub.S({dot over (V)}.sub.DC-{dot
over (V)}.sub.Proof)[A] [22]
i.sub.sense={dot over
(x)}.gradient.C.sub.S(V.sub.DC-V.sub.proof)+C.sub.S({dot over
(V)}.sub.DC-{dot over (V)}.sub.Proof).apprxeq.{dot over
(x)}.gradient.C.sub.S(V.sub.DC-V.sub.Proof) [23]
[0105] Given the symmetric nature of the drive design, the left and
right sense currents are 180.degree. out of phase with each other,
as shown in equations 24 and 25. This lends itself well to the
fully-differential closed-loop architecture shown in FIG. 7.
i SL = x . .gradient. C SL ( V DC - V B ) = - x . C o g o ( V DC -
V Proof ) [ 24 ] i SR = x . .gradient. C SR ( V DC - V B ) = + x .
C o g o ( V DC - V Proof ) [ 25 ] ##EQU00009##
[0106] The first-stage amplifier converts this transduced current
of equations 24 and 25 into a voltage signal. The voltage at the
output terminals of the amplifier (non-inverting/inverting,
respectively) is given by equations 26 and 27.
V + = V DC + I SL Z F = V DC - j.omega..DELTA. x C o g o ( V DC - V
Proof ) Z F .ident. V DC - V AC [ 26 ] V - = V DC + I SR Z F = V DC
+ j.omega..DELTA. x C o g o ( V DC - V Proof ) Z F .ident. V DC + V
AC [ 27 ] ##EQU00010##
[0107] The feedback impedance transfer function, Z.sub.F,
determines the gain and phase lag introduced into the closed loop
drive as shown in equation 28. The goal is to provide an adequate
gain with minimal imposed phase lag.
Z F ( .omega. ) = R F 1 + j.omega. R F C F [ .OMEGA. ] [ 28 ]
##EQU00011##
[0108] For charge amplifier (i.e., current integrator) C-to-V drive
loop designs, (R.sub.FC.sub.F).sup.-1<<.omega..sub.o. For
transimpedance (i.e., current to voltage) I-to-V designs,
(R.sub.FC.sub.F).sup.-1>>.omega..sub.o.
[0109] A secondary gain stage provides an additional signal boost
(.alpha.) along with a required signal inversion. As a result, a
positive AC voltage change will pull the proof mass in the +x
direction. Effectively, the output signals following the secondary
stage are oriented such that the detected sense current provides
the necessary reinforcing drive behavior (see FIG. 7).
[0110] A comparator is used to extract a timing reference signal
(i.e., a "sync" trigger) used to coordinate the processing of
timing events in one or more of the cosine, arcsine, arccosine, and
arctangent algorithms. For drive designs using a transimpedance
amplifier, the timing reference signal tracks proof mass velocity
because sense current is directly proportional the rate of change
of MEMS displacement (see equation 23).
[0111] Examination of equation 19 reveals that applied force is
proportional to both the AC and DC drive voltage levels. This
suggests that one can linearly control the force by manipulating
either signal variable (or both). The method described here makes
use of the DC drive level to effect automatic control of the
displacement amplitude.
[0112] A digital proportional-integral-derivative (PID) controller
compares an active measurement of proof mass displacement amplitude
(obtained by one or more of the cosine, arcsine, arccosine, and
arctangent algorithms) to a desired setpoint level to produce an
error signal. The PID controller determines a computed drive
voltage level based on the error signal. With the appropriate
setting of the PID coefficients, feedback action drives the
steady-state error signal to zero thus enforcing automatic
regulation of the displacement amplitude.
[0113] In general, for steady-state oscillation, the loop must
satisfy the Barkhausen stability criteria, which are necessary by
not sufficient conditions for stability. The Barkhausen criteria
require, first, that the magnitude of the loop gain, |T(j.omega.)|,
is equal to unity and, second, that the phase shift around the loop
is zero or an integer multiple of 2.pi.: .angle.T(j.omega.)=2.pi.n,
n.epsilon.0, 1, 2 . . . .
[0114] Using the closed-loop transfer functions summarized in FIG.
8, one can determine the DC drive voltage requirement for stable
oscillation (i.e., |T(j.omega..sub.o)|=1), as shown in equation
29.
V DC = m .omega. o 2 Q R F .gradient. C S .gradient. C D .alpha. +
V Proof [ 29 ] ##EQU00012##
[0115] The electronics can induce a phase shift which can move the
oscillation frequency away from the desired mechanical resonance
frequency. A transimpedance amplifier will cause a phase lag which
produces a negative frequency shift as shown in equations 30 and
31.
.DELTA. .theta. S = - tan - 1 ( .omega. R F C F ) [ rad ] [ 30 ]
.omega. * = positive root { .omega. 2 + .omega. .omega. Q tan (
.DELTA. .theta. S - .pi. 2 ) - .omega. o 2 } [ rad sec ] [ 31 ]
##EQU00013##
[0116] In comparison, a charge amplifier produces a positive
frequency shift resulting from its induced phase lead, as shown in
equation 32.
.DELTA. .theta. S = .pi. 2 - tan - 1 ( .omega. R F C F ) [ 32 ]
##EQU00014##
[0117] The gain loss is a measure of the degradation of the
mechanical displacement resulting from a phase-shifted isolation
frequency .omega.* as shown in equation 33 and 34.
Gain Loss = 20 Log 10 ( H ( j.omega. * ) H ( j.omega. o ) ) [ 33 )
H ( s ) = L { x x } = 1 s 2 + s .omega. 0 Q + .omega. 0 2 [ 34 ]
##EQU00015##
[0118] FIG. 10 depicts a Bode plot with a magnitude graph 1000 and
a phase graph 1050. The magnitude graph 1000 depicts the magnitude
of the transfer function of equation 34 at various quality factors
Q. The phase graph 1050 depicts the phase of the transfer function
of equation 34 as a function of frequency for various quality
factors Q.
[0119] FIG. 11 depicts a graph 1100 showing the change in
oscillation frequency as a function of phase shift for various
quality factors Q. As the quality factor decreases, the oscillation
frequency exhibits a greater dependence on the phase shift.
[0120] FIG. 12 depicts a graph 1200 showing the gain loss as a
function of phase shift. As the quality factor Q increases, the
gain loss becomes more dependent on phase shift. Thus, for a given
phase shift, low-Q resonators demonstrate a larger shift in
oscillation frequency with minimal loss of displacement amplitude.
The contrary is true for high-Q devices; gain loss is more severe
while oscillation frequency deviation remains small.
[0121] FIG. 13 depicts a graph 1300 illustrating a start-up
procedure of a resonating proof mass (e.g., 102, 203, 608 (FIGS. 1,
2, and 6)). The resonator's start-up procedure includes a sequence
of open-loop high-voltage pulses that are alternately applied to
left and right drive capacitors (e.g., 114a and 114b, respectively
(FIG. 1)) at the expected resonant frequency of the proof mass
(e.g., 102, 203, 608 (FIGS. 1, 2, and 6)). In some examples, the
pulses are applied with a duty cycle of approximately 20-25%. The
graph 1300 includes a left comb drive voltage curve 1302 and a
right comb drive voltage curve 1304. The graph 1300 also includes a
time 1306 at which the open-loop start-up sequence ends and the
resonator is oscillated in closed-loop drive mode at a lower
voltage. The graph 1300 also includes a time interval 1308
depicting the settling time of the resonator, corresponding to the
time required for the resonator to enter into steady-state
oscillations. In the example depicted in FIG. 13, the settling time
interval 1308 is approximately 10 milliseconds from initiation of
the start-up sequence. As depicted in FIG. 13, an initial set of
open-loop high-voltage drive pulses are used to rapidly increase
the amplitude of the oscillation of the proof mass. The left and
right comb drive voltage curves 1302 and 1304 include alternating
pulses, so that the proof mass (e.g., 102, 203, 608 (FIGS. 1, 2,
and 6)) is driven for a greater proportion of the time. The
symmetric pulse depicted in FIG. 13 rapidly drives the proof mass
(e.g., 102, 203, 608 (FIGS. 1, 2, and 6)) from rest to an amplitude
close to the desired steady-state amplitude prior to engaging the
digital closed-loop controller. In the example depicted in FIG. 13,
after an initial set of twenty pulses per drive comb (e.g., 114a,
114b (FIG. 1)), the digital closed-loop controller is engaged to
regulate the displacement amplitude of the proof mass (e.g., 102,
203, 608 (FIGS. 1, 2, and 6)) to the desired level. The number of
required pulses and rate are predetermined for a given inertial
device (e.g., 100, 202, 602 (FIGS. 1, 2, and 6)). Pulse rate can be
adjusted based on temperature and/or previously measured data, such
as resonant frequency.
[0122] FIG. 14 depicts a graph 1400 showing sense signals of an
inertial device (e.g., 100, 202, 602 (FIGS. 1, 2, and 6)) during
resonator start-up. The graph 1400 includes a drive sense signal
102, a drive velocity curve 1404, a proof mass displacement curve
1406, and a sense current curve 1408. The graph 1400 also includes
a time 1410 at which the open-loop start-up ends and the digital
closed-loop controller is engaged. The time 1410 corresponds to the
time 1306 (FIG. 13). The open-loop sequence prior to the time 1410
lasts approximately 7 milliseconds and the proof mass (e.g., 102,
203, 608 (FIGS. 1, 2, and 6)) experiences an approximately linear
increase in amplitude with time. After the time 1410, closed-loop
regulation maintains the desired displacement amplitude and rejects
disturbances. By applying alternating high-voltage open-loop pulses
prior to engaging the digital closed-loop controller, the proof
mass (e.g., 102, 203, 608 (FIGS. 1, 2, and 6)) is rapidly brought
to the desired oscillation amplitude.
[0123] Once the initial start-up sequence is complete and the
displacement of the proof mass (e.g., 102, 203, 608 (FIGS. 1, 2,
and 6)) is sufficiently large to produce timing measurements for
use in determining inertial perimeters, a TDS algorithm (e.g., one
or more of the cosine, arcsine, arccosine, and arctangent
algorithms) can determine amplitude estimates of the proof mass
displacement. A digital proportional-integral-derivative (PID)
scheme can provide effective closed-loop regulation of resonator
oscillation. The digital controller (e.g., 736, 818 (FIGS. 7 and
8)) is derived from an equivalent continuous time PID controller
design. The digital controller (e.g., 736, 818 (FIGS. 7 and 8))
accepts digital displacement measurements that are subtracted from
a desired setpoint to produce an error signal, which is processed
to produce an updated drive DC voltage setting (V.sub.DC). The
drive force is proportional to the DC and AC drive voltage levels.
The DC drive voltage is controlled using a digital potentiometer
and a buffer circuit. The controller's maximum update rate is set
by the sample rate, which is in turn determined by the resonant
frequency of the proof mass (e.g., 102, 203, 608 (FIGS. 1, 2, and
6)). The digital controller (e.g., 736, 818 (FIGS. 7 and 8))
determines the DC drive voltage according to equation 35 below.
v DC ( t ) = K p e ( t ) + K i .intg. 0 t e ( .tau. ) .tau. + K d t
e ( t ) [ V ] [ 35 ] ##EQU00016##
[0124] Where e(t) is the difference between the desired setpoint
and the measured displacement.
[0125] The Laplace transform of equation 35 is shown in equation
36.
V DC ( s ) = [ K p + K i s + s K d ] E ( s ) [ 36 ]
##EQU00017##
[0126] For digital implementations, the Z-transform is more
appropriate and is shown in equation 37.
V DC ( z ) = [ K p + K i 1 - z - 1 + ( 1 - z - 1 ) K d ] E ( z ) [
37 ] ##EQU00018##
[0127] Equation 37 can be rearranged as shown in equation 38.
V DC ( z ) = [ ( K p + K i + K d ) - ( K p + 2 K d ) z - 1 + K d z
- 2 ] 1 - z - 1 E ( z ) [ 38 ] ##EQU00019##
[0128] The digital PID coefficients can be defined in terms of the
more intuitive continuous-time coefficients as shown in equations
39, 40, and 41.
K.sub.1=K.sub.p+K.sub.i+K.sub.d [39]
K.sub.2=-K.sub.p-2K.sub.d [40]
K.sub.3=K.sub.d [41]
[0129] Equations 39, 40, and 41 can be used to rewrite equation 38
as shown in equation 42.
V.sub.DC(z)-z.sup.-1V.sub.DC(z)=[K.sub.1+K.sub.2z.sup.-1+K.sub.3z.sup.-2-
]E(z) [42]
[0130] Equation 42 can be converted to a difference equation
suitable for implementation as shown in equation 43.
v.sub.DC(n)=v.sub.DC(n-1)+K.sub.1e(n)+K.sub.2e(n-1)+K.sub.3e(n-2)
[43]
[0131] The digital error signal e(n) in equation 43 represents the
difference between the desired displacement and displacement
amplitude measurements, as shown in equation 44.
e(n)=.DELTA.x.sub.Setpoint-.DELTA.x.sub.n[m] [44]
[0132] The block diagram 800 (FIG. 8) can be used to determine a
V.sub.DC gain setting required to stabilize the closed-loop drive
when the digital loop converges to steady state as shown in
equation 45 and 46.
( T ( j.omega. ) = 1 ) , [ 45 ] V DC = m .omega. o 2 Q R F
.gradient. C S .gradient. C D .alpha. + V Proof [ 46 ]
##EQU00020##
[0133] Selection of PID perimeters can be performed either manually
or automatically using additional algorithms to obtain adequate
enclosed-loop performance. When properly tuned, a good regulator
design should provide a favorable balance between robust stability
and rejection of disturbances in the resonator's displacement
amplitude.
[0134] FIG. 15 depicts a system 1500 that includes two TDS
structures and a graph 1550 that depicts capacitance profiles of
the TDS structures. The system 1500 includes an in-phase TDS
structure 1501 and an out-of-phase TDS structure 1503. The system
1500 includes a fixed beam 1502 and a movable beam 1504. The
in-phase TDS structure 1501 includes movable teeth 1510a, 1510b,
and 1510c (collectively, movable teeth 1510) on the movable beam
1504 and fixed teeth 1506a, 1506b, and 1506c (collectively, fixed
teeth 1506) on the fixed beam 1502. The out-of-phase TDS structure
1503 includes movable teeth 1512a, 1512b, 1512c, and 1512d
(collectively, movable teeth 1512) on the movable beam 1504 and
fixed teeth 1508a, 1508b, and 1508c (collectively, fixed teeth
1508) on the fixed beam 1502. The fixed beam 1502 is connected by
an anchor (not shown) to a top layer (not shown) and/or a bottom
layer (not shown). The movable beam 1504 is connected by a spring
1514a to an anchor 1516a and by a spring 1514b to an anchor 1516b.
The movable beam 1504 oscillates as shown by the arrow 1505. As the
movable beams 1504 oscillates, the capacitance of the TDS
structures 1501 and 1503 varies. FIG. 15 depicts pitch distances
1518a, 1518b, 1518c, and 1518d, all of which are equal, and any of
which can be referred to as the pitch distance 1518.
[0135] The graph 1550 illustrates the variation of capacitance of
the TDS structures 1501 and 1503 with displacement of the movable
beam 1504. The graph 1550 includes capacitance curves 1552 and 1554
corresponding to capacitance of the TDS structures 1501 and 1503,
respectively. The graph 1550 includes a pitch distance 1556,
corresponding to the pitch distance 1518. The graph 1550 includes
displacement levels 1556, 1558, 1560, 1562, 1564, 1566, 1568, 1570,
1572, 1574, 1576, 1578, and 1580, spaced by one-fourth the pitch
distance 1556. Because of the movable teeth 1512 are offset by
one-half the pitch distance 1518 from the fixed teeth 1508 when the
movable beam 1504 is in the rest position, while the movable teeth
1510 are aligned with the fixed teeth 1506 when the movable beam
1504 is in the rest position, the capacitance curve 1554 is
180.degree. out of phase from the capacitance curve 1552. The
out-of-phase capacitance curve 1554 can be subtracted from the
in-phase capacitance curve 1552 to generate a differential
capacitance signal. The quantity d.sub.0 is defined by equation
47.
d 0 .ident. Pitch 2 [ 47 ] ##EQU00021##
[0136] FIG. 16 depicts a system 1600 for determining acceleration
with a TIA. The system 1600 includes an inertial device 1602 which
can include any or all of the features of the inertial devices
described herein (e.g., 100, 202, 602 (FIGS. 1, 2, and 6)). The
inertial device 1602 includes a proof mass 1604 which can include
any or all of the features of the proof masses as described herein
(e.g., 102, 203, 608 (FIGS. 1, 2, and 6)). The inertial device 1602
includes an in-phase TDS structure 1606 and an out-of-phase TDS
structure 1608. The TDS structures 1606 and 1608 can include any
and all the features of the TDS structures described herein (e.g.,
105, 207, 506, 604, 606, 1501, 1503 (FIGS. 1, 2, 5, 6, and 15)).
The TDS structures 1606 and 1608 produce analog output signals 1626
and 1628, respectively. The analog output signals 1626 and 1628 are
received by a differential transimpedance amplifier 1610. The
differential transimpedance amplifier 1610 includes an operational
amplifier 1638, feedback resistors 1632 and 1636 and feedback
capacitors 1634 and 1630. The values of the feedback capacitor 1630
and the feedback resistor 1632 are selected to satisfy equation 48,
where .omega..sub.0 is the oscillation frequency of the proof mass
1604.
1 R F C F .omega. .sigma. [ 48 ] ##EQU00022##
[0137] The values of the feedback capacitor 1634 and the feedback
resistor 1636 are also chosen to satisfy equation 48. The
transimpedance amplifier 1610 generates an in-phase TIA output
signal 1612 and an out-of-phase TIA output signal 1613. The TIA
output 1612 and 1613 are received by a low-pass filter 1614 which
removes higher-frequency components to generate respective analog
signals 1616 and 1617. The analog signals 1615 and 1617 are
received by a comparator 1616 which compares the two signals and
generates a rectangular-wave signal 1618 with pulse edges
corresponding to times at which a difference of the output signals
1612 and 1613 crosses zero.
[0138] The rectangular-wave signal 1618 is received by a TDC 1620
which generates digital timestamps of rising and falling edges of
the rectangular-wave signal 1618. The TDC 1620 receives a sync
signal 1622 comprising a sync pulse and uses the sync pulse as an
index for determining the timestamps. The timestamps generated by
the TDC 1620 are received by digital circuitry 1624 which
implements the cosine algorithm to determine inertial parameters,
including acceleration of the inertial device 1602. By using the
differential transimpedance amplifier 1610 to measure differential
signals of the inertial device 1602, the system 1600 can reject
common-mode noise.
[0139] FIG. 17 depicts a graph 1700 illustrating signals of the
system 1600 (FIG. 16). The graph 1700 includes a displacement curve
1702, an in-phase TIA output curve 1704 and an out-of-phase TIA
output curve 1706. The displacement curve 1702 illustrates the
motion of the proof mass 1604 with respect to time. As the proof
mass 1604 oscillates, the in-phase TIA output curve 1704 and the
out-of-phase TIA output curve 1706 also oscillate and cross each
other. The graph 1700 includes an pulse signal 1708 that represents
the rectangular-wave signal 1618 generated by the comparator
1616.
[0140] The graph 1700 includes points 1710, 1712, 1714, 1716, 1718,
1720, 1722 and 1724, each corresponding to an integral multiple of
the quantity d.sub.0, which is defined as one-half of the pitch
distance of the TDS structures 1606 and 1608 (FIG. 16). As the
graph 1700 illustrates, the in-phase TIA output curve 1704 crosses
the out-of-phase TIA output curve 1706, and the pulse signal 1708
changes value, when the proof mass 1604 (FIG. 16) is at distances
from its rest position that are equal to integer multiples of
d.sub.0. Thus, the motion of the proof mass 1604 (FIG. 16) can be
determined from crossings of the TIA output curves 1704 and 1706
independently of bias voltage applied to the TDS structures 1606
and 1608 (FIG. 16).
[0141] FIG. 18 depicts a system 1800 for determining acceleration
with a CA. The system 1800 includes an inertial device 1802 which
can include any or all of the features of the inertial devices
described herein (e.g., 100, 202, 602 (FIGS. 1, 2, and 6)). The
inertial device 1802 includes a proof mass 1804 which can include
any or all of the features of the proof masses as described herein
(e.g., 102, 203, 608 (FIGS. 1, 2, and 6)). The inertial device 1802
includes an in-phase TDS structure 1806 and an out-of-phase TDS
structure 1808. The TDS structures 1806 and 1808 can include any
and all the features of the TDS structures described herein (e.g.,
105, 207, 506, 604, 606, 1501, 1503, 1606, 1608 (FIGS. 1, 2, 5, 6,
15, and 16)). The TDS structures 1806 and 1808 produce analog
output signals 1826 and 1828, respectively. The analog output
signals 1826 and 1828 are received by a differential charge
amplifier 1810. The differential charge amplifier 1810 includes an
operational amplifier 1838, feedback resistors 1832 and 1836 and
feedback capacitors 1834 and 1830. The values of the feedback
capacitor 1830 and the feedback resistor 1832 are selected to
satisfy equation 49, where coo is the oscillation frequency of the
proof mass 1804.
1 R F C F .omega. .sigma. [ 49 ] ##EQU00023##
[0142] The values of the feedback capacitor 1834 and the feedback
resistor 1836 are also chosen to satisfy equation 49. The charge
amplifier 1810 generates an in-phase TIA output signal 1812 and an
out-of-phase TIA output signal 1813. The TIA output 1812 and 1813
are received by a low-pass filter 1814 which removes
higher-frequency components to generate respective analog signals
1816 and 1817. The analog signals 1815 and 1817 are received by a
comparator 1816 which compares the two signals and generates a
rectangular-wave signal 1818 with pulse edges corresponding to
times at which a difference of the output signals 1812 and 1813
crosses zero.
[0143] The rectangular-wave signal 1818 is received by a TDC 1820
which generates digital timestamps of rising and falling edges of
the rectangular-wave signal 1818. The TDC 1820 receives a sync
signal 1822 comprising a sync pulse and uses the sync pulse as an
index for determining the timestamps. The timestamps generated by
the TDC 1820 are received by digital circuitry 1824 which
implements the cosine algorithm to determine inertial parameters,
including acceleration of the inertial device 1802. By using the
differential charge amplifier 1810 to measure differential signals
of the inertial device 1802, the system 1800 can reject common-mode
noise.
[0144] FIG. 19 depicts a graph 1900 illustrating signals of the
system 1800 (FIG. 18). The graph 1900 includes a displacement curve
1902, an in-phase TIA output curve 1904 and an out-of-phase TIA
output curve 1906. The displacement curve 1902 illustrates the
motion of the proof mass 1804 (FIG. 18) with respect to time. As
the proof mass 1804 (FIG. 18) oscillates, the in-phase TIA output
curve 1904 and the out-of-phase TIA output curve 1906 also
oscillate and cross each other. The graph 1900 includes a pulse
signal 1908 that represents the rectangular-wave signal 1818 (FIG.
18) generated by the comparator 1816 (FIG. 18).
[0145] The graph 1900 includes points 1910, 1912, 1914, 1916, 1918,
1920, 1922 and 1924, each corresponding to a proof mass
displacement that satisfies the quantity d.sub.0(n+1/2), where
d.sub.0 is defined as one-half of the pitch distance of the TDS
structures 1806 and 1808 (FIG. 18) and n is an integer. As the
graph 1900 illustrates, the in-phase TIA output curve 1904 crosses
the out-of-phase TIA output curve 1906, and the pulse signal 1908
changes value, when the proof mass 1804 (FIG. 18) is at distances
from its rest position that satisfy the quantity d.sub.0(n+1/2).
Thus, the motion of the proof mass 1804 (FIG. 18) can be determined
from crossings of the TIA output curves 1904 and 1906 independently
of bias voltage applied to the TDS structures 1806 and 1808 (FIG.
18).
[0146] When a charge amplifier is used instead of a transimpedance
amplifier, the circuit topology is the same but the feedback time
constant of the charge amplifier is selected such that its
frequency pole is placed at frequencies much lower than the
resonant frequency of the proof mass (e.g., 102, 203, 608, 1604,
1804 (FIGS. 1, 2, 6, 16, and 18)), as shown in equation 49. This
ensures that the motion-induced current is integrated to produce an
output that is proportional to the time-varying capacitance of the
TDS structures (e.g., 105, 207, 506, 604, 606, 1501, 1503, 1606,
1608, 1806, 1808 (FIGS. 1, 2, 5, 6, 15, 16, and 18)). As the time
integral of current is charge, integrating the motion-induced
current can be also described as charge amplification.
[0147] In some examples, a TDS structure (e.g., 105, 207, 506, 604,
606, 1501, 1503, 1606, 1608, 1806, 1808 (FIGS. 1, 2, 5, 6, 15, 16,
and 18)) can be coupled to a mechanical oscillator to generate a
time-varying periodic capacitive signal. The period of the
capacitive signal is based on the geometry of the TDS structure,
which is fixed by the fabrication process. This enables an inertial
device to have stability and insensitivity to variations in
amplitude and frequency of the oscillator. The time-varying
capacitive signal is measured due to determine inertial parameters
such as acceleration and rotation of the inertial device. In some
examples, discrete-time systems and methods such as switched
capacitor amplifiers and commutating, or chopping, modulating
techniques are used to measure the time-varying capacitive signal.
In some examples, a bridge with a general impedance converter (GIC)
can be used. In some examples, continuous-time architectures such
as a transimpedance amplifier or a charge amplifier can be
used.
[0148] In some examples, an inertial device (e.g., 100, 202, 602,
1602, 1802 (FIGS. 1, 2, 6, 16, and 18)) including a proof mass
(e.g., 102, 203, 608, 1604, 1804 (FIGS. 1, 2, 6, 16, and 18)) and a
TDS structure (e.g., 105, 207, 506, 604, 606, 1501, 1503, 1606,
1608, 1806, 1808 (FIGS. 1, 2, 5, 6, 15, 16, and 18)) can measure
position of the proof mass by converting a motion-induced
capacitive current from the TDS structure to a voltage using the
TIA circuit architecture depicted in FIG. 16.
[0149] The total capacitive current (e.g., 1626, 1628 (FIG. 16)) is
determined by the time-derivative of the charge of the relevant
capacitor (e.g., TDS structures, 105, 207, 506, 604, 606, 1501,
1503, 1606, 1608, 1806, 1808 (FIGS. 1, 2, 5, 6, 15, 16, and 18)).
Capacitor charge (Q) is the product of capacitance (C) and voltage
potential across the capacitor (V.sub.C) as shown in equation
50.
{dot over (q)}=i= V.sub.C+C{dot over (V)}.sub.C [50]
[0150] If series resistance is approximately zero, an operational
amplifier provides a virtually fixed potential across the
capacitor, then the right-most term in equation 50, which includes
the first time derivative of the capacitor voltage, can be
neglected, resulting in equation 51.
i .apprxeq. C . V C = C t V C = C x x t V C = C x x . V C [ 51 ]
##EQU00024##
[0151] Therefore, the capacitive current (e.g., 1626, 1628 (FIG.
16)) is approximately equal to the product of the gradient of the
physical capacitor (dC/dx), the velocity of the proof mass (i), and
the fixed potential across the capacitor (V.sub.C). In some
examples, the capacitor design can include structures that force
the capacitive gradient (dC/dx) to zero at geometrically fixed
locations. Thus, as long as the voltage across the capacitor
(V.sub.C) is not zero and the proof mass (e.g., 102, 203, 608,
1604, 1804 (FIGS. 1, 2, 6, 16, and 18)) has a non-zero velocity at
these locations, at least some of the times at which the capacitive
current (i) is zero correspond to times at which the proof mass
passes these geometrically fixed locations. This enables the
zero-crossings of the capacitive current to be used in determining
that the proof mass has either crossed the zero-gradient locations
or has come to rest (such as at minimum or maximum displacements).
The slope of the current signal (i) is given by its time derivative
as shown in equation 52.
i t = C V C = t { C x x t V C } = t { C x } x t V C + C x x V C [
52 ] ##EQU00025##
[0152] Equation 52 can be rewritten as equation 53.
i t = ( 2 C x 2 x . 2 + C x x ) V C [ 53 ] ##EQU00026##
[0153] As a result, the rate of change of the capacitive current
(e.g., 1626, 1628 (FIG. 16)) is proportional to the voltage across
the capacitor (V.sub.C), the curvature of the spatial capacitance
(d.sup.2C/dx.sup.2), and the square of the proof mass velocity
({dot over (x)}.sup.2). It is also in proportion to the gradient
(dC/dx) of the capacitance and the acceleration of the proof mass
({dot over (x)}). Typically, the peak curvature of the capacitance
occurs when the proof mass is near a position in which the
capacitive gradient approaches zero and this coincides with a zero
acceleration and maximum velocity condition. Therefore, equation 53
can be approximated as shown in equation 54.
i t .apprxeq. 2 C x 2 x . 2 V C [ 54 ] ##EQU00027##
[0154] An accurate measurement of the time associated with a
zero-crossing of the current can be used to determine acceleration
of the inertial device (e.g., 100, 202, 602, 1602 and 1802). These
zero-crossing times correspond to fixed physical displacements of
the proof mass (e.g., 102, 203, 608, 1604, 1804 (FIGS. 1, 2, 6, 16,
and 18)) located at integer multiples of one-half of the pitch,
where the pitch is the periodic spacing of the teeth in the TDS
structure (e.g., 105, 207, 506, 604, 606, 1501, 1503, 1606, 1608,
1806, 1808 (FIGS. 1, 2, 5, 6, 15, 16, and 18)). FIG. 15 depicts an
example of differential TDS structures 1501 and 1503 having a pitch
1518.
[0155] Uncertainty in the measurement of the zero-crossing times is
given by the ratio of the electronic noise amplitude to the rate of
the signal crossing. Therefore, maximizing the slope of the current
signal (di/dt) can minimize the timing uncertainty of the
zero-crossings of the current. Larger values of the capacitive
curvature (d.sup.2C/dx.sup.2) velocity of the proof mass ({dot over
(x)}) and bias by voltage (V.sub.C) can increase the slope of the
current signal (di/dt) and thus would reduce uncertainty in the
time measurements. These parameters can be selected based on a
desired rate of signal crossing as well as other parameters to
achieve a desired performance of the inertial device (e.g., 100,
202, 602, 1602, 1802 (FIGS. 1, 2, 6, 16, and 18)). A differential
TIA can be implemented as depicted in FIG. 16. The time constant of
the feedback impedance (e.g., 1630, 1632, 1634, 1636) sets the
bandwidth of the TIA 1610. The design of the TIA 1610 can balance
several competing variables which can influence the performance of
the sensor. For example, a large feedback resistance (e.g., 1632,
1636) improves the signal gain, but also reduces bandwidth and
reduces noise. The bandwidth must be large enough to pass the
spectral content of the capacitive signal (e.g., 1626, 1628)
without attenuation and frequency-dependent phase shift. Signal
attenuation will reduce signal slope which in turn will increase
the measurement uncertainty of the zero-crossing. Excessive phase
shift can distort the measured times leading to harmonic distortion
of the output signals (e.g., 1612, 1613) of the TIA 1610.
[0156] In some examples, an inertial device (e.g., 100, 202, 602,
1602, 1802 (FIGS. 1, 2, 6, 16, and 18)) can include signal
processing circuitry such as a charge amplifier that generates an
output voltage that is proportional to the time-varying capacitance
signal of a TDS structure (e.g., 105, 207, 506, 604, 606, 801,
1503, 1606, 1608, 1806, 1808). This architecture for signal
processing circuitry can be referred to as a capacitance-to-voltage
(C-to-V) architecture.
[0157] In some examples, bridge circuits may be used to determine
times at which a proof mass (e.g., 102, 203, 608, 1604, 1804 (FIGS.
1, 2, 6, 16, and 18)) is at positions at which the capacitance of
structures (e.g., 105, 207, 506, 604, 606, 1501, 1503, 1606, 1608,
1806, 1808 (FIGS. 1, 2, 5, 6, 15, 16, and 18)) have equivalent
values (i.e., become balanced). In some examples, this balancing
may occur when in-phase TDS structures (e.g., 604, 1501, 1606,
1806) have equivalent capacitance values as out-of-phase TDS
structures (e.g., 606, 1503, 1608, 1808). For an example, as
depicted in the graph 1550 of FIG. 15, the in-phase capacitance
curve 1552 crosses the out-of-phase capacitance curve 1554 at odd
multiples of displacement increments of one-fourth of the
pitch.
[0158] One example of a C-to-V signal processing circuit is a
charge amplifier (e.g., 1810). The charge amplifier 1810 shares the
same circuit topology as the transimpedance amplifier 1610, but the
charge amplifier 1810 has a pole location that is at very low
frequencies. The pole location is determined by the time constant
of the feedback impedance (e.g., 1830, 1832, 1834, 1836). The
charge amplifier 1810 has a low frequency pole occurring at a
frequency much lower than the mechanical resonant frequency of the
proof mass (e.g., 102, 203, 608, 1604, 1804 (FIGS. 1, 2, 6, 16, and
18)). The current amplifier 1810 can operate to integrate current,
which is operationally equivalent to charge amplification. In some
examples, the pole placement is selected to be much lower than the
mechanical resonant frequency of the proof mass (e.g., 102, 203,
608, 1604, 1804 (FIGS. 1, 2, 6, 16, and 18)) and sufficiently low
so as not to impose phase shift on the time-varying capacitive
signal (e.g., 1826, 1828). The ratio of bias voltage (set by
operational amplifier common mode voltage) to the feedback
capacitor value determines the gain (i.e., volts per Farad), of the
output signal (e.g., 1812, 1813). A smaller value of feedback
capacitance (e.g., 1830, 1834) and/or a larger bias voltage will
increase the signal gain. In some examples, noise is largely
determined by the value of the feedback capacitor (e.g., 1830,
1834), especially if this value is quite small. The feedback
capacitor's contribution to output RMS noise is proportional to
sqrt(kT/C.sub.F). Other factors such as operational amplifier
current and voltage noise density, parasitic capacitance, and
low-pass filtering also affect the signal-to-noise ratio.
[0159] By measuring times at which the output (e.g., 1826, 1828) of
the differential charge amplifier (e.g., 1810) crosses zero, the
inertial parameters of the inertial device (e.g., 100, 202, 602,
1602, 1802 (FIGS. 1, 2, 6, 16, and 18)) can be determined. Because
the differential TDS structures (e.g., 1806, 1808) are well-matched
with a 180.degree. phase relationship, they will provide reliable
crossing events that correspond to fixed physical displacements.
When using a charge amplifier, timing uncertainty is minimized by
maximizing the capacitor gradients (i.e. first spatial derivative
of the capacitance) occurring at odd multiples of one-fourth of the
pitch of the TDS structure (e.g., 105, 207, 506, 604, 606, 1501,
1503, 1606, 1608, 1806, 1808 (FIGS. 1, 2, 5, 6, 15, 16, and 18)).
In some examples, an inertial device (e.g., 100, 202, 602, 1602,
1802 (FIGS. 1, 2, 6, 16, and 18)) can have a proof mass of (e.g.,
102, 203, 608, 1604, 1804) with a 1.2 kHz resonant frequency and a
drive displacement amplitude of 12 microns. If this inertial device
experiences a 1 g DC acceleration and 0.5 g and 0.2 g sinusoidal
inputs at 2.5 Hz and 33.1 Hz respectively, the inertial device can
have a noise level of approximately 10 .mu.g/rtHz.
[0160] As described with reference to FIG. 6, inertial parameters
can be determined from zero-crossing times by using transimpedance
amplifiers, charge amplifiers, or switched capacitors to convert
analog input signals to analog voltages. In some cases a comparator
and a TDC can be used to measure zero-crossing times as digital
inputs to digital circuitry implementing zero-crossing times as
digital inputs to digital circuitry implementing a cosine
algorithm. In some examples an ADC can be used in place of the
comparator and TDC to digitize the analog voltages and to produce
time measurements in the digital domain. Digital signal processing
techniques, such as upsampling and interpretation, may be used to
enhance the accuracy of ADC zero-crossing detection.
[0161] The methods and systems described herein utilize the
periodic nature of the motion of a proof mass (e.g., 102, 203, 608,
1604, 1804 (FIGS. 1, 2, 6, 16, and 18)), in conjunction with TDS
structures (e.g., 105, 207, 506, 604, 606, 1501, 1503, 1606, 1608,
1806, 1808 (FIGS. 1, 2, 5, 6, 15, 16, and 18)) that generate
capacitive signals (e.g., 1626, 1628, 1826, 1828) with measureable
timing events. The timing events can be zero-crossings and can
correspond to fixed spatial locations at known multiples of
half-pitch and/or quarter-pitch spacing of the TDS structure. The
underlying mathematical formulations behind the methods assume that
the displacement offset due to an input acceleration of the
inertial device of (e.g., 100, 202, 602, 1602, 1802 (FIGS. 1, 2, 6,
16, and 18)) is constant over the duration of the resonant period,
and that the sinusoidal proof mass motion is spectrally pure. In
practice, the methods rely on approximations that assume that the
input offset is quasi-static, exhibiting change much slower than
the time scale of the resonant period. If the input offset changes
more rapidly, this can result in an increased harmonic distortion
of output signals.
[0162] The cosine algorithm can be implemented as described below,
but can also be implemented with other forms of timing intervals,
each having implications for noise, frequency response, and
harmonic distortion performance.
[0163] The proof mass (e.g., 102, 203, 608, 1604, 1804 (FIGS. 1, 2,
6, 16, and 18)) can be modeled as a simple harmonic oscillator
experiencing a quasi-static input acceleration, and the
steady-state displacement response of the proof mass takes the form
of a sinusoidal signal with displacement amplitude (.DELTA.x) and
constant offset (.DELTA.d) as shown in equation 55.
x(t)=.DELTA.xcos(.theta.(t))+.DELTA.d [55]
[0164] In these examples, quasi-static displacement accelerations
are inertial excitations evolving over time scales much longer than
the resonant period of oscillation (i.e., frequency is approaching
the zero). In these examples, the relationship between input
acceleration ({umlaut over (x)}) and physical offset is represented
by equation 56.
.DELTA. d .apprxeq. L - 1 { lim s .fwdarw. 0 H ( s ) L { x } } = L
- 1 { lim s .fwdarw. 0 L { x } s 2 + s .omega. o Q + .omega. o 2 }
= x .omega. o 2 [ 56 ] ##EQU00028##
[0165] FIG. 20 depicts a graph 2000 illustrating displacement of a
proof mass and TDS timing events. The graph 2000 includes a
displacement curve 2002 corresponding to displacement of the proof
mass (e.g., 102, 203, 608, 1604, 1804 (FIGS. 1, 2, 6, 16, and 18)).
The graph 2000 also includes timing events 2004, 2006, 2008 and
2010 that correspond to threshold crossing times, which can be
determined using the systems and methods described with reference
to FIGS. 4, 6, 15, 16, 17, 18, and 19. The graph 2000 also includes
time intervals 2012, 2014, 2024 and 2026, and these points 2016,
2018, 2020 and 2022 used in the systems and method described
herein.
[0166] In general the inertial device (e.g., 100, 202, 602, 1602,
1802 (FIGS. 1, 2, 6, 16, and 18)) provides measurements of times at
which the proof mass crosses four known physical location (e.g.,
2004, 2006, 2008, and 2010) as dictated by the geometry of the TDS
structure (e.g., 105, 207, 506, 604, 606, 1501, 1503, 1606, 1608,
1806, 1808 (FIGS. 1, 2, 5, 6, 15, 16, and 18)). These known
physical locations correspond to a specific spatial phase points
(e.g., 2016, 2018, 2020 and 2024). The measured time intervals
(e.g., times at which the proof mass is at 2004, 2006, 2008 and
2010) can be used to determine an input acceleration of the
inertial device. Under the quasi-static assumption, the input
acceleration can be directly determined while eliminating
dependence on displacement amplitude (.DELTA.x) and frequency. This
is because information about the period of the oscillation of the
proof mass is actively measured each cycle by measuring the time
intervals 2024, 2026. The time intervals are measured with respect
to the peak of the cosine carrier signal or oscillation of the
proof mass, and the static input offset assumption ensures symmetry
about the peak. Equations 57, 58, 59, and 60 can be used to express
the relationships between physical locations, phase points,
measured time intervals and oscillation offset.
.DELTA.xcos(.theta..sub.1)+.DELTA.d=.DELTA.xcos(2.pi.f.sub.ot.sub.1)+.DE-
LTA.d=d.sub.1 [57]
.DELTA.xcos(.theta..sub.2)+.DELTA.d=.DELTA.xcos(2.pi.f.sub.ot.sub.2)+.DE-
LTA.d=d.sub.2 [58]
.DELTA.xcos(.theta..sub.3)+.DELTA.d=.DELTA.xcos(2.pi.f.sub.ot.sub.3)+.DE-
LTA.d=d.sub.3 [59]
.DELTA.xcos(.theta..sub.4)+.DELTA.d=.DELTA.xcos(2.pi.f.sub.ot.sub.4)+.DE-
LTA.d=d.sub.4 [60]
[0167] The resonant frequency is inversely related to the resonant
period and can be measured independently using time intervals 2024
and 2026 as shown in equation 61.
f o = 1 T 1 = 1 T 2 [ 61 ] ##EQU00029##
[0168] The average resonant frequency estimated can be determined
from the average measured period as shown in equation 62.
f avg = 1 T avg = 2 T 1 + T 2 [ 62 ] ##EQU00030##
[0169] In equation 63, equations 59 and 58 are added and
trigonometric sum-to-product formulas are applied.
2.DELTA.d+.DELTA.xcos(2.pi.f.sub.ot.sub.3)+.DELTA.xcos(2.pi.f.sub.ot.sub-
.2)=2.DELTA.d+2.DELTA.xcos(2.pi.f.sub.o(t.sub.3+t.sub.2))cos(.pi.f.sub.o.d-
elta..sub.32)=d.sub.3+d.sub.2 [63]
[0170] In equation 64, equations 60 and 57 are added and
trigonometric sum-to-product formulas are applied.
2.DELTA.d+.DELTA.xcos(2.pi.f.sub.ot.sub.4)+.DELTA.xcos(2.pi.f.sub.ot.sub-
.1)=2.DELTA.d+2.DELTA.xcos(2.pi.f.sub.o(t.sub.4+t.sub.1))cos(.pi.f.sub.o.d-
elta..sub.41)=d.sub.4+d.sub.1 [64]
[0171] By subtracting equation 64 from equation 63 and
incorporating equation 61, the dependence on offset (.DELTA.d) can
be eliminated and an expression for the displacement amplitude can
be obtained as shown in equation 65. The subscript indicates the
current measurement cycle.
d 3 + d 2 - d 4 - d 1 = 2 .DELTA. x [ cos ( .pi. T 2 ( t 3 + t 2 )
) cos ( .pi. T 2 .delta. 32 ) - cos ( .pi. T 1 ( t 4 + t 1 ) ) cos
( .pi. T 1 .delta. 41 ) ] .thrfore. .DELTA. x n = ( d 3 + d 2 - d 4
- d 1 2 ) 1 cos ( .pi. T 2 ( t 3 + t 2 ) ) cos ( .pi. T 2 .delta.
32 ) - cos ( .pi. T 1 ( t 4 - t 1 ) ) cos ( .pi. T 1 .delta. 41 ) [
65 ] ##EQU00031##
[0172] By adding equations 63 and 64 and substituting the
expression for displacement amplitude, an expression for
displacement offset ((.DELTA.d) can be obtained as shown in
equation 66.
.DELTA. d = d 1 + d 2 + d 3 + d 4 4 - .DELTA. x 2 [ cos ( .pi. T 2
( t 3 + t 2 ) ) cos ( .pi. T 2 .delta. 32 ) + cos ( .pi. T 1 ( t 4
+ t 1 ) ) cos ( .pi. T 1 .delta. 41 ) ] [ 66 ] ##EQU00032##
[0173] Equation 66 can be rearranged as shown in equation 67.
.thrfore. .DELTA. d = d 1 + d 2 + d 3 + d 4 4 - ( d 3 + d 2 - d 4 -
d 1 4 ) [ cos ( .pi. T 2 ( t 3 + t 2 ) ) cos ( .pi. T 2 .delta. 32
) + cos ( .pi. T 1 ( t 4 + t 1 ) ) cos ( .pi. T 1 .delta. 41 ) cos
( .pi. T 2 ( t 3 + t 2 ) ) cos ( .pi. T 2 .delta. 32 ) - cos ( .pi.
T 1 ( t 4 + t 1 ) ) cos ( .pi. T 1 .delta. 41 ) ] [ 67 ]
##EQU00033##
[0174] Using the approximate input-output relationship between
acceleration and displacement derived in equation 56, an expression
for sensed input acceleration, scaled to units of g, can be derived
as shown in equation 68.
x n = ( 2 .pi. f avg ) 2 g ( d 1 + d 2 + d 3 + d 4 4 - ( d 3 + d 2
- d 4 - d 1 4 ) [ cos ( .pi. T 2 ( t 3 + t 2 ) ) cos ( .pi. T 2
.delta. 32 ) + cos ( .pi. T 1 ( t 4 + t 1 ) ) cos ( .pi. T 1
.delta. 41 ) cos ( .pi. T 2 ( t 3 + t 2 ) ) cos ( .pi. T 2 .delta.
32 ) - cos ( .pi. T 1 ( t 4 + t 1 ) ) cos ( .pi. T 1 .delta. 41 ) ]
) [ 68 ] ##EQU00034##
[0175] In some examples, it may be difficult to accurately measure
time intervals using a peak of a displacement curve as a reference
point. In some examples, this issue can be overcome by taking
advantage of several aspects. First, the output of TDS structures
(e.g., 105, 207, 506, 604, 606, 1501, 1503, 1606, 1608, 1806, 1808
(FIGS. 1, 2, 5, 6, 15, 16, and 18)) includes measurable
zero-crossings for every traversal of a proof mass (e.g., 102, 203,
608, 1604, 1804 (FIGS. 1, 2, 6, 16, and 18)) past a given spatial
reference point (d.sub.0). Second, the timing intervals for
zero-crossings corresponding to the traversal of the same spatial
reference point are symmetric about the maxima and minima of the
displacement curve or the analog output signal (e.g., 611, 613,
1626, 1628, 1826, 1828 (FIGS. 6, 16, and 18)). Therefore, the
timing intervals used in the cosine method (e.g., T.sub.1, T.sub.2,
T.sub.3, and T.sub.4 of equations 61-68) are one-half of these
timing intervals between zero-crossings. As described herein, the
d.sub.0 spatial reference points can include teeth of the TDS
structures (105, 207, 506, 604, 606, 1501, 1503, 1606, 1608, 1806,
1808 (FIGS. 1, 2, 5, 6, 15, 16, and 18)). In some examples, the
motion-induced current produced by the TDS structure (e.g., 105,
207, 506, 604, 606, 1501, 1503, 1606, 1608, 1806, 1808 (FIGS. 1, 2,
5, 6, 15, 16, and 18)) is measured using a transimpedance
amplifier, and d.sub.0 corresponds to the first-half pitch spacing
(e.g., for a pitch=4.5 .mu.m count d.sub.0=2.25 .mu.m) of the teeth
of the TDS structure. It some examples, a capacitance-to-voltage or
charge-amplifier conversion device would produce zero-crossings
that correspond to odd multiples of the quarter-pitch spacing of
the TDS structure. Fabrication-induced variations can modify the
effective d.sub.o scale factor used in implementing the cosine
algorithm. This is accounted for with a small calibration
correction.
[0176] In some examples, an inertial device (e.g., 100, 202, 602,
1602, 1802 (FIGS. 1, 2, 6, 16, and 18)) can include a TDC to
provide high-resolution measurements of zero-crossing times with
respect to a rising edge of a sync signal generated based on a
drive-sync signal (which has a 90.degree. lead and is proportional
to the velocity of a proof mass (e.g., 102, 203, 608, 1604, 1804
(FIGS. 1, 2, 6, 16, and 18)) of the inertial device. The TDC can
have programmable masking windows and edge sensitivity to enable
extracting timing of specific events as depicted in FIG. 21.
[0177] FIG. 21 depicts a graph 2100 showing various time intervals
that can be extracted from a differential TIA output. The graph
2100 includes a proof mass displacement curve 2102 illustrating the
position of a proof mass (e.g., 102, 203, 608, 1604, 1804 (FIGS. 1,
2, 6, 16, and 18)) over time. The graph 2100 also includes a
differential TIA output curve 2104 illustrating an analog output
(e.g., 624, 626, 1612, 1613 (FIGS. 6 and 16)) of a TIA (e.g., 620,
1610 (FIGS. 6 and 16)) receiving an analog signal (e.g., 611, 613,
1626, 1628 (FIGS. 6 and 16)) from the proof mass. The graph 2100
also includes a comparator output curve 2108 that illustrates the
digital output of a comparator (e.g., 1620 (FIG. 16)). As a digital
signal, the comparator output curve 2108 consists substantially of
two values: a high value and a low value. The comparator output
curve 2108 transitions from a low value to a high value when the
TIA output curve 2102 crosses zero and has a positive slope. The
comparator output curve 2108 transitions from a high value to a low
value when the TIA output curve 2014 crosses zero and has a
negative slope. Thus, the comparator output curve 2108 is a digital
representation of the zero-crossing times of the TIA output curve
2104.
[0178] The graph 2100 includes reference levels 2112 and 2114, each
corresponding to a displacement of magnitude d.sub.0, or one-half
the pitch distance, of the proof mass from its neutral position.
Because a displacement of magnitude d.sub.0 results in a minimum in
capacitance (for an in-phase TDS structure such as 1606 (FIG. 16))
or a maximum in capacitance (for an out-of-phase TDS structure such
as 1608 (FIG. 16)), the spatial capacitance gradient and thus the
capacitive current is zero. Thus, the TIA output curve 2104 crosses
zero when the proof mass displacement curve 2102 crosses either of
the reference levels 2112 or 2114. The graph 2100 includes times
2116, 2122, 2124, and 2130 at which the displacement curve 2102
crosses the reference level 2114 and times 2118, 2120, 2126, and
2128 at which the displacement curve 2102 crosses the reference
level 2112. The graph 2100 also includes time intervals 2132, 2134,
2136, and 2138. The time interval T.sub.1 2132 corresponds to the
time interval between the times 2116 and 2124. The time interval
T.sub.2 2134 corresponds to the time interval between the times
2118 and 2126. The time interval T.sub.32 2136 corresponds to the
time interval between the times 2126 and 2128. The time interval
T.sub.412138 corresponds to the time interval between the times
2124 and 2130. The graph 2100 also includes a digital sync signal
2142. In some examples, this can be a sync signal created from a
drive sync signal. The sync time interval 2140 corresponds to the
time interval between adjacent rising edges of the digital sync
signal 2142. The TDC (e.g., 1620 (FIG. 16)) can use the most recent
rising edge of the digital sync signal 2142 as a reference for
determining digital timestamps of changes in value of the
comparator output signal 2108. These timestamps can be used to
determine the time intervals 2132, 2134, 2136, and 2138.
[0179] In summary, the data determined per cycle includes a sync
time 2140 marking the interval between the previous sync rising
edge and the most recent zero-crossing times (e.g., 2116, 2118,
2120, 2122) measured with respect to the most recent rising edge of
the digital sync signal 2108. One reason the sync time 2140 is
useful is for establishing a relationship with past timing events
so that time intervals crossing the sync boundaries (e.g., 2132,
2134) can be computed for zero-crossing and period
measurements.
[0180] Equations 69-78 can be used to compute output acceleration
at time n.
d 1 = d 4 = - d o [ 69 ] d 2 = d 3 = + d o [ 70 ] T 32 = t 3 n - t
2 n [ 71 ] T 41 = t 4 n - t 1 n [ 72 ] .delta. 41 = .delta. 32 = 0
[ 73 ] .theta. 1 = .theta. 4 = .omega. o T 41 2 = .pi. T 41 T 1 [
74 ] .theta. 2 = .theta. 3 = .omega. o T 32 2 = .pi. T 32 T 2 [ 75
] T 1 = t sync n - t 1 n - 1 + t 1 n [ 76 ] T 2 = t sync n - t 2 n
- 1 + t 2 n [ 77 ] f avg = 1 T avg = 2 T 1 + T 2 [ 78 ]
##EQU00035##
[0181] Equations 69-78 can be substituted directly into equations
65 and 68 to produce measurements of displacement amplitude and
output acceleration one or more times per oscillation cycle of the
proof mass (e.g., 102, 203, 608, 1604, 1804 (FIGS. 1, 2, 6, 16, and
18)). A set of simultaneous equations can be written for each
reference level (e.g., 2112, 2114) as shown in equations 79 and
80.
.DELTA. x cos ( .pi. T 32 T 2 ) + x .omega. o 2 = + d o [ 79 ]
.DELTA. x cos ( .pi. T 41 T 1 ) + x .omega. o 2 = - d o [ 80 ]
##EQU00036##
[0182] Equations 79 and 80 can be subtracted to solve for
displacement amplitude at the n.sup.th measurement cycle as shown
in equation 81.
.DELTA. x n = 2 d o cos ( .pi. T 41 T 1 ) - cos ( .pi. T 32 T 2 ) [
81 ] ##EQU00037##
[0183] Similarly, equations 79 and 80 can be added and the
expression for displacement amplitude substituted to solve for
input acceleration for the inertial device (e.g., 100, 202, 602,
1602, 1802 (FIGS. 1, 2, 6, 16, and 18)), scaled to units of g as
shown in equation 82.
x n = d o ( 2 .pi. f avg ) 2 g cos ( .pi. T 41 T 1 ) + cos ( .pi. T
32 T 2 ) cos ( .pi. T 41 T 1 ) - cos ( .pi. T 32 T 2 ) [ 82 ]
##EQU00038##
[0184] One reason the cosine algorithm is useful is that it is
relatively insensitive to variations in the amplitude and frequency
of the oscillation of the proof mass. Since the cosine algorithm
rejects variations in these parameters for timescales much longer
than the resonant period of the oscillation, the cosine algorithm
generally has excellent drift performance.
[0185] The low-drift performance of the systems and methods
described herein is significantly attributable to the fact that the
timing measurements ultimately relate back to a fixed, known
reference dimension (i.e., the pitch of the TDS structures such as
105, 207, 506, 604, 606, 1501, 1503, 1606, 1608, 1806, 1808 (FIGS.
1, 2, 5, 6, 15, 16, and 18)). Recalling that d.sub.0 is defined as
one-half the pitch, equations 81 and 82 can be rewritten as shown
in equations 83 and 84 to isolate the terms involving timing
measurement to a ratio that itself defines a ratio of the desired
output variable to the geometrically fixed pitch of the TDS
structure.
.DELTA. x n Pitch = 1 cos ( .pi. T 41 T 1 ) - cos ( .pi. T 32 T 2 )
[ 83 ] x n Pitch = ( 2 .pi. f avg ) 2 2 g cos ( .pi. T 41 T 1 ) +
cos ( .pi. T 32 T 2 ) cos ( .pi. T 41 T 1 ) - cos ( .pi. T 32 T 2 )
[ 84 ] ##EQU00039##
[0186] Estimating output acceleration noise performance given
simulated front-end electronic noise and signal properties (i.e.,
the slope of the zero-crossings) can be useful for quickly
evaluating candidate inertial sensor designs and iterating the
design process to achieve desired performance specifications
without requiring detailed, time-consuming simulations. The
following illustrates a derivation of a simple equation which can
be used to evaluate output noise density (g/rtHz). The derivation
begins with equation 82, applies perturbation analysis by
considering the measured time intervals with additive white
Gaussian noise (i.e., jitter, E), and includes some simplifying
assumptions. Equations 85-90 illustrate the addition of white
Gaussian noise to equation 82.
T ^ 32 = T 32 + .epsilon. 32 [ 85 ] T ^ 41 = T 41 + .epsilon. 41 [
86 ] T ^ 1 = T 1 + .epsilon. P 1 [ 87 ] T ^ 2 = T 2 + .epsilon. P 2
[ 88 ] T ^ avg = T ^ 1 + T ^ 2 2 = T 1 + T 2 2 + .epsilon. T 1 +
.epsilon. T 2 2 = T avg + .epsilon. _ T [ 89 ] x n = d o g ( 2 .pi.
T ^ avg ) 2 cos ( .pi. T ^ 41 T 1 ) + cos ( .pi. T ^ 32 T ^ 2 ) cos
( .pi. T ^ 41 T ^ 1 ) - cos ( .pi. T ^ 32 T ^ 2 ) [ 90 ]
##EQU00040##
[0187] Equation 91 is an approximation of equation 90 based noting
that the noise terms of the expansion of {circumflex over
(T)}.sub.avg.sup.2 are small compared to T.sub.avg.sup.2.
x n = d o ( 2 .pi. ) 2 g 1 T avg 2 + 2 T avg .epsilon. _ T +
.epsilon. _ T 2 cos ( .pi. T ^ 41 T ^ 1 ) + cos ( .pi. T ^ 32 T ^ 2
) cos ( .pi. T ^ 41 T ^ 1 ) - cos ( .pi. T ^ 32 T ^ 2 ) .apprxeq. d
o g ( 2 .pi. T avg ) 2 cos ( .pi. T ^ 41 T ^ 1 ) + cos ( .pi. T ^
32 T ^ 2 ) cos ( .pi. T ^ 41 T ^ 1 ) - cos ( .pi. T ^ 32 T ^ 2 ) [
91 ] ##EQU00041##
[0188] The common trigonometric sum-difference formula shown in
equation 92 can be used to expand noise terms as shown in equation
93.
cos ( .alpha. + .beta. ) = cos ( .alpha. ) cos ( .beta. ) - sin (
.alpha. ) sin ( .beta. ) [ 92 ] x n .apprxeq. d o g ( 2 .pi. T avg
) 2 cos ( .pi. T 41 T 1 + .epsilon. T 1 ) cos ( .pi. .epsilon. 41 T
1 + .epsilon. T 1 ) - sin ( .pi. T 41 T 1 + .epsilon. T 1 ) sin (
.pi. .epsilon. 41 T 1 + .epsilon. T 1 ) + cos ( .pi. T 32 T 2 +
.epsilon. T 2 ) cos ( .pi. .epsilon. 32 T 2 + .epsilon. T 2 ) - sin
( .pi. T 32 T 2 + .epsilon. T 2 ) sin ( .pi. .epsilon. 32 T 2 +
.epsilon. T 2 ) cos ( .pi. T 41 T 1 + .epsilon. T 1 ) cos ( .pi.
.epsilon. 41 T 1 + .epsilon. T 1 ) - sin ( .pi. T 41 T 1 +
.epsilon. T 1 ) sin ( .pi. .epsilon. 41 T 1 + .epsilon. T 1 ) - cos
( .pi. T 32 T 2 + .epsilon. T 2 ) cos ( .pi. .epsilon. 32 T 2 +
.epsilon. T 2 ) + sin ( .pi. T 32 T 2 + .epsilon. T 2 ) sin ( .pi.
.epsilon. 32 T 2 + .epsilon. T 2 ) [ 93 ] ##EQU00042##
[0189] The cosine terms involving the timing jitter (.epsilon.) are
small with zero mean so the small-angle approximation can be
applied as shown in equation 94.
x n .apprxeq. d o g ( 2 .pi. T ^ avg ) 2 cos ( .pi. T 41 T 1 +
.epsilon. T 1 ) cos ( .pi. .epsilon. 41 T 1 + .epsilon. T 1 ) - sin
( .pi. T 41 T 1 + .epsilon. T 1 ) sin ( .pi. .epsilon. 41 T 1 +
.epsilon. _ T 1 ) + cos ( .pi. T 32 T 2 + .epsilon. _ T 2 ) cos (
.pi. .epsilon. 32 T 2 + .epsilon. _ T 2 ) - sin ( .pi. T 32 T 2 +
.epsilon. _ T 2 ) sin ( .pi. .epsilon. 32 T 2 + .epsilon. T 2 ) cos
( .pi. T 41 T 1 + .epsilon. T 1 ) cos ( .pi. .epsilon. 41 T 1 +
.epsilon. T 1 ) - sin ( .pi. T 41 T 1 + .epsilon. T 1 ) sin ( .pi.
.epsilon. 41 T 1 + .epsilon. T 1 ) - cos ( .pi. T 32 T 2 +
.epsilon. T 2 ) cos ( .pi. .epsilon. 32 T 2 + .epsilon. T 2 ) + sin
( .pi. T 32 T 2 + .epsilon. T 2 ) sin ( .pi. .epsilon. 32 T 2 +
.epsilon. T 2 ) [ 94 ] ##EQU00043##
[0190] The cosine terms involving the timing jitter (.epsilon.) are
small with zero mean so the small-angle approximation can once
again be applied. The multiplicative sine terms involving the
measured periods (T.sub.41 and T.sub.32) are, at their largest,
bounded by .+-.1. This will be a convenient consideration when
examining the variance of {umlaut over (x)}.sub.n, and this also
causes the noise estimation to be somewhat conservative. Because
the period measurements and the denominator of each sinusoidal
argument are much larger than the noise, the noise term here can be
ignored. Furthermore, the resulting noise terms in denominator are
negligible compared to the cosine terms. Applying all of these
approximations to equation 94, the individual terms of the
resulting expression can be collected into two main terms as shown
in equation 95, one representing the approximate expected value of
the output acceleration (left term), and the other containing noise
error components (right term).
x n .apprxeq. d o g ( 2 .pi. T avg ) 2 cos ( .pi. T 41 T 1 ) - cos
( .pi. T 32 T 2 ) cos ( .pi. T 41 T 1 ) - cos ( .pi. T 32 T 2 ) + d
o g ( 2 .pi. T avg ) 2 .pi. .epsilon. 41 T 1 + .pi. .epsilon. 32 T
2 cos ( .pi. T 41 T 1 ) - cos ( .pi. T 32 T 2 ) = x n _ + noise [
95 ] ##EQU00044##
[0191] The cosine terms in the denominator of the noise portion of
equation 95 can be replaced with the expression in equation 96
involving the displacement amplitude to result in equation 97.
1 cos ( .pi. T 41 T 1 ) - cos ( .pi. T 32 T 2 ) = .DELTA. x n 2 d o
[ 96 ] x n .apprxeq. x n _ + d o g ( 2 .pi. T avg ) 2 .pi.
.epsilon. 41 T 1 + .pi. .epsilon. 32 T 2 cos ( .pi. T 41 T 1 ) -
cos ( .pi. T 32 T 2 ) = x n _ + .DELTA. x n 2 g ( 2 .pi. T avg ) 2
( .pi. .epsilon. 41 T 1 + .pi. .epsilon. 32 T 2 ) [ 97 ]
##EQU00045##
[0192] Another simplification can be made because T.sub.avg,
T.sub.1, and T.sub.2 are approximately equal, as shown in equation
98.
x n .apprxeq. x n _ + .DELTA. x n 4 g ( 2 .pi. T avg ) 3 (
.epsilon. 41 + .epsilon. 32 ) [ 98 ] ##EQU00046##
[0193] Taking the variance of {umlaut over (x)}.sub.n allows the
computation of total noise power (note:
.omega..sub.o=2.pi./T.sub.avg) as shown in equation 99.
.sigma. x n 2 .apprxeq. E [ ( .DELTA. x n 4 g .omega. o 3 (
.epsilon. 41 + .epsilon. 32 ) ) 2 ] [ g RMS 2 ] [ 99 ]
##EQU00047##
[0194] The jitter properties associated with individual time
measurements (t.sub.1, t.sub.2, t.sub.3, t.sub.4) are assumed to be
uncorrelated. As a consequence the measured interval timing jitter
(.epsilon..sub.41 and .epsilon..sub.32) is also uncorrelated.
Further, it is assumed that the jitter variance of each timing
event is identical (.sigma..sub.t1=.sigma..sub.t2= . . .
=.sigma..sub.t, this is approximately true and sufficient for a
noise estimation) as shown in equation 100. Using these arguments
we conclude,
.sigma..sub..epsilon..sub.41=.sigma..sub..epsilon..sub.32=.sigma..sub..ep-
silon.= {square root over (2)}.sigma..sub.t.
.sigma. x n .apprxeq. 0.85 .DELTA. x n 4 g .omega. o 3 2 .sigma.
.epsilon. = 0.85 .DELTA. x n 2 g .omega. o 3 .sigma. t [ g RMS ] [
100 ] ##EQU00048##
[0195] The final noise equation, illustrated in equation 100,
includes the product of displacement magnitude (.DELTA.x.sub.n[m]),
the cube of the resonance frequency
(.omega..sub.0.sup.3[rad.sup.3/sec.sup.3]), and the timing event
jitter (.sigma..sub.t [sec.sub.RMS]). The additional factor of 0.85
is an empirical factor that calibrates the estimation with both
laboratory and detailed simulation results.
[0196] Edge timing jitter (.sigma..sub.t) can be readily estimated
from the total integrated electronics noise divided by the signal
slope at the zero-crossing event. Simulations may be used to
provide values for the electronic noise and signal slew rate.
Differential or single-ended noise and slews may be used as long as
the consistency is observed. TDC timing uncertainty may also be
included by root-sum-squaring with the electronics induced jitter
as shown in equation 101.
.sigma. t = Total Integrated Electronics Noise [ V RMS ] Slope at
Zero Crossing [ V / sec ] [ sec RMS ] [ 101 ] ##EQU00049##
[0197] The one-sided TDS output acceleration noise density may be
estimated using equation 100 and dividing the results by the
square-root of the sensor bandwidth (i.e., Nyquist sampling
rate=Bandwidth=f.sub.o/2), giving equation 102. When using the
double-sampled TDS cosine algorithm, where one applies the cosine
algorithm once for the positive and once for the negative half of
the resonant period (thus producing two samples each cycle), one
should substitute twice the bandwidth (i.e., bandwidth=f.sub.o).
Equation 102 can be used for noise performance estimation and
compares well with experimental results.
.eta. x n .apprxeq. 0.85 .DELTA. x n 2 g .omega. o 3 .sigma. t
Bandwidth [ g RMS Hz ] [ 102 ] ##EQU00050##
[0198] In some examples, an ADC may be used to convert analog
outputs (e.g., 626, 1615, 1617, 1815, 1817 (FIGS. 6, 16, and 18))
of AFE's (e.g., 616) such as charge amplifiers (e.g., 618, 1810
(FIGS. 6 and 18)) or transimpedance amplifiers (e.g., 620, 1610
(FIGS. 6 and 16)) to digital representations (e.g., 635 (FIG. 6)).
The digital representations (e.g., 635 (FIG. 6)) can be received by
digital circuitry that can determine timestamps of threshold
crossings and can implement the cosine algorithm of equations 81
and 82 to determine inertial parameters (e.g., acceleration and
proof mass displacement) of the inertial device (e.g., 100, 202,
602, 1602, 1802 (FIGS. 1, 2, 6, 16, and 18)).
[0199] FIG. 22 depicts a summing block 2200 illustrating signal
flows for using an ADC to digitally reproduce an analog input
signal. The summing block 2200 includes input differential
capacitance signals 2202 that can be received from differential TDS
structures (e.g., 105, 207, 506, 604, 606, 1501, 1503, 1606, 1608,
1806, 1808 (FIGS. 1, 2, 5, 6, 15, 16, and 18)). The input
differential capacitance signals 2202 can be signals such as any of
the signals 1611, 1613, 1626, 1628, 1826, 1828 (FIGS. 16 and 18).
The differential capacitance signals 2202 are received by an AFE
2204. The AFE 2204 can be a charge amplifier or transimpedance
amplifier. The ADC 2204 introduces ADC input noise 2208, the
injection of which is schematically represented by the summing
block 2206. The output of the summing block 2206 is received by a
low-pass filter 2210 which removes higher frequency components of
the signal. The output of the low-pass filter 2210 is received by
an ADC 2212. The ADC 2212 performs sample-and-hold 2214 at a
sampling rate of 400 kS/sec. The ADC 2212 also includes a quantizer
2218. The ADC 2212 produces a digital output signal 2220 that is a
digital representation of the difference between the input
differential capacitor signals 2202. Digital circuitry can receive
the digital signal 2220 and perform upsampling, interpolation, and
further post-processing to determine inertial parameters.
[0200] The digital circuitry receiving the digital signal 2220 can
extract inertial information using one or more of interpolation,
trigonometric functions, and inverse trigonometric functions. An
example of a trigonometric function that the digital circuitry can
implement is the cosine function. Examples of inverse trigonometric
functions that the digital circuitry can implement include the
arcsine, arccosine, and arctangent functions.
[0201] Interpolation can be used to improve the timing accuracy of
threshold crossing times of the digital signal 2220 measured by the
digital circuitry. The digital circuitry can interpolate and
upsample the digital signal 2220 to produce a higher-resolution
indication of threshold crossing by the digital signal 2220. The
interpolation can include linear interpolation and/or splined
interpolation.
[0202] The digital circuitry can use a sync signal that is derived
from a drive signal or a drive sense signal to synchronize measured
threshold crossings with known positions of the proof mass (e.g.,
102, 203, 608, 1604, 1804 (FIGS. 1, 2, 6, 16, and 18)). Because the
sync signal is derived from the drive or the drive sense signal,
there is a known phase relationship between the sync signal and the
position of the proof mass. In some examples, the drive signal is
90.degree. out-of-phase with respect to the proof mass motion, and
a phase lead or lag can be applied as necessary to properly align
the sync signal. In some examples, the sync signal is not required
to be exactly aligned to the digital signal (e.g., 2220). The
required alignment accuracy is related to the time between zero
crossings of the digital signal (e.g., 2220). As long as the sync
signal is sufficiently aligned to fall within the correct interval
between zero-crossings of the digitized signal, the position of the
proof mass within its oscillation can be determined with sufficient
accuracy.
[0203] The digital circuitry can use a cosine function to determine
displacement and/or acceleration as follows. The digital circuitry
can determine threshold crossings of the interpolated and upsampled
digital signal 2220. The digital circuitry can then determine time
intervals between the threshold crossings, and can determine
quantities containing ratios of the time intervals. The digital
circuitry can then determine results of cosine functions of these
quantities, as illustrated in equations 81 and 82. The digital
circuitry can implement the cosine method as described with
reference to FIGS. 1-21, except that the threshold crossings can be
detected by the digital circuitry instead of by a comparator and
TDC.
[0204] The digital circuitry can use an inverse trigonometric
function to determine displacement of the proof mass (e.g., 102,
203, 608, 1604, 1804 (FIGS. 1, 2, 6, 16, and 18)) as follows. The
digital circuitry can apply a trigonometric function or other
periodic function to a quantity comprising a ratio of the proof
mass displacement to the periodicity of the teeth of a TDS
structure (e.g., 105, 207, 506, 604, 606, 1501, 1503, 1606, 1608,
1806, 1808 (FIGS. 1, 2, 5, 6, 15, 16, and 18)). The periodicity of
the teeth can be the pitch of the arrays of the teeth. By
determining an inverse of the determined trigonometric function,
the digital circuitry can extract the ratio of the displacement to
the periodicity. Because the periodicity is a known constant that
is determined during fabrication of inertial device, subsequent
determination of the inertial parameters can be obtained by
multiplying the extracted ratio by the periodicity. The periodic
function need not be a trigonometric function; for
non-trigonometric functions, an appropriate inverse function is
used. In general, the digital circuitry can extract a ratio of the
displacement of the proof mass (e.g., 102, 203, 608, 1604, 1804
(FIGS. 1, 2, 6, 16, and 18)) to any physical position or dimension,
and the function does not need to be periodic. When the function is
periodic, the ratio of displacement to periodicity is proportional
to the phase of the periodic function. Changes in displacement
induced by accelerations of the inertial device (e.g., 100, 202,
602, 1602, 1802 (FIGS. 1, 2, 6, 16, and 18)) will result in phase
shifts of the periodic function.
[0205] FIG. 23 depicts the use of linear interpolation to determine
zero crossings of a digitized signal (e.g., 2220 (FIG. 22)). FIG.
23 includes three views 2300, 2330, and 2360, each depicting
successively enlarged views of a threshold crossing. The view 2300
includes a displacement curve 2302 depicting the motion of a proof
mass (e.g., 102, 203, 608, 1604, 1804 (FIGS. 1, 2, 6, 16, and 18))
and a differential ADC output curve 2304, which can be the digital
signal 2220. The view 2300 also includes a threshold 2310 which, in
this example, occurs at the 0 V level, and an area of interest 2320
in which the ADC output curve 2304 crosses the threshold 2310.
However, the measurement of the time at which the ADC output curve
2304 crosses the threshold 2310 is limited by the time resolution
of the ADC unless further processing is performed. Without further
processing, all that can be determined is that the ADC output curve
2304 crossed the threshold 2310 at some time between two samples of
the ADC.
[0206] The view 2330 is an enlarged view of the area of interest
2320 and depicts the use of interpolation to improve the resolution
of the zero-crossing time measurement. The view 2330 includes
points 2334 and 2336, which are points sampled by the ADC and are
thus points on the ADC output curve 2304. The point 2334 is below
the threshold 2310, while the point 2336 is above the threshold
2310. Thus, the ADC output curve 2304 crossed the threshold 2310
sometime between point 2334 and point 2336. With the time and
voltage values of the points 2334 and 2336 known, linear
interpolation can be performed to determine the time at which a
straight line 2332 drawn between the points 2334 and 2336 would
intersect the threshold 2310. In some examples, splined or
polynomial interpolation is performed to determine the time at
which a curved line drawn between the points 2334 and 2336 would
intersect the threshold 2310. This intersection is illustrated in
the view 2330 by a point 2338. The point 2338 is the digitally
interpolated estimate of the threshold crossing time of the analog
signal represented by the digital ADC output curve 2304. The view
2330 also includes an area of interest 2350 centered on the point
2338.
[0207] The view 2360 is an enlarged view of the area of interest
2350 and depicts the error incurred by digital interpolation. The
view 2336 includes the curve 2364 which is the analog signal
represented by the ADC output curve 2304. The analog curve 2334
crosses the threshold 2310 at a true crossing point 2368. The time
interval between the digitally estimated point 2338 and the true
crossing point 2368 is represented by time interval 2372. The time
interval 2372 is thus the error of the timing measurement obtained
by digital interpolation. Because the time interval 2372 is smaller
than the sampling rate of the ADC, the interpolation has improved
the accuracy and resolution of the threshold crossing time
measurements.
[0208] FIG. 24 depicts a graph 2400 illustrating zero-crossings of
an ADC digital output signal. The graph 2400 includes a
displacement curve 2402 representing displacement of a proof mass
(e.g., 102, 203, 608, 1604, 1804 (FIGS. 1, 2, 6, 16, and 18)) of an
inertial device (e.g., 100, 202, 602, 1602, 1802 (FIGS. 1, 2, 6,
16, and 18)). The graph 2400 includes a differential AFE output
curve 2404 that represents an output of an AFE (e.g., 626 (FIG.
6)). The graph 2400 also includes an ADC output curve 2404
representing a digital output (e.g., 2220 (FIG. 22)) of an ADC
(e.g., 2212 (FIG. 22)). The ADC output curve 2406 is a digital
representation of the AFE output curve 2404. The graph 2400 depicts
points 2411, 2413, 2415, and 2417 on the displacement curve 2402
that correspond to times at which the proof mass crosses reference
points that are multiples of d.sub.0 from the neutral position. The
differential AFE output curve 2404 includes points 2412, 2414,
2416, and 2418 that are threshold crossings corresponding to the
points 2411, 2413, 2415, and 2417, but with a time delay. The time
delay is imposed by analog band-limiting filtering.
[0209] FIG. 25 depicts a graph 2500 that shows an upsampled AFE
output curve. The graph 2500 includes the displacement curve 2402
(FIG. 24), the differential AFE output curve 2404 (FIG. 24), and
the points 2411, 2413, 2415, and 2417 (FIG. 24). The graph 2500
also includes an upsampled AFE output curve 2506. The output curve
2506 has been upsampled to a sampling rate triple that of the AFE
output curve 2406 (FIG. 24). The upsampled output curve 2506 has a
lower timing uncertainty in zero-crossing measurements, which
results in better timing resolution. Digital upsampling and
interpolation can eliminate spurious artifacts for a given input
noise level and can provide better agreement between the ADC output
curve 2506 and the AFE output curve 2404.
[0210] Digital interpolation and zero-crossing detection in the
digital domain can result in performance equivalent to analog
zero-crossing detection. As described herein, digital
interpolation, upsampling, and filtering a band-limited input
signal can produce a higher-resolution rendition of the original
input signal. This improves the accuracy of subsequent digital
zero-crossing detection. In digital zero-crossing detection,
digital circuitry detects when a digital signal crosses zero and
applies local linear interpolation to determine a precise crossing
time. In some examples, the digital circuitry may apply hysteresis
to reduce the effects of signal noise at the decision boundaries.
These digital systems and methods can result in accuracy at least
as high as analog zero-crossing systems and methods.
[0211] The arccosine algorithm and the arcsine algorithm can be
used by digital processing circuitry to determine displacement
information from a periodic nonlinear signal. Both the arccosine
algorithm and the arcsine algorithm operate on the output of an
ADC. The difference between the arccosine and arcsine algorithms is
that the arccosine algorithm is implemented on signals generated by
arrays of teeth that have opposing teeth that are aligned in the
neutral position, while the arcsine algorithm is implemented on
analog signals generated by arrays of teeth that have teeth that
are offset by one-fourth of the tooth pitch (or a spatial phase
shift of 90.degree.) at the neutral position.
[0212] In some examples, the arccosine algorithm can be implemented
using arrays of teeth designed for the arcsine algorithm by
including a 90.degree. spatial phase shift. Conversely, the arcsine
algorithm can operate on signals generated by arrays of teeth
designed for the arccosine algorithm by including a 90.degree.
spatial phase shift. Signals generated by arrays of teeth with
arbitrary offsets in the neutral position can also be operated on
by either the arccosine or the arcsine algorithm by including an
appropriate spatial phase shift corresponding to the offset. For
teeth that are offset by a phase .phi. in the neutral position, the
orthogonal capacitance signals can be described by equations
103-105.
C I ( t ) .apprxeq. A cos ( 2 .pi. P x ( t ) + .PHI. ) [ 103 ] C Q
( t ) .apprxeq. A sin ( 2 .pi. P x ( t ) + .PHI. ) [ 104 ] C Q ( t
) C I ( t ) = tan ( 2 .pi. P x ( t ) + .PHI. ) [ 105 ]
##EQU00051##
[0213] Where displacement is described by equation 106, the offset
phase .phi. can also be expressed as an effective offset to x(t) as
shown in equation 107.
x ( t ) = A sin ( w 0 t ) + x Inertial ( t ) [ 106 ] C I ( t )
.apprxeq. A cos ( 2 .pi. P ( x ( t ) + x 0 ) ) [ 107 ]
##EQU00052##
[0214] The quantity x.sub.0 is defined by equation 108.
x 0 = .PHI. P 2 .pi. [ 108 ] ##EQU00053##
[0215] The arccosine and arcsine algorithms can be implemented
using only one of the two signals C.sub.I and C.sub.Q. Thus, by
measuring C.sub.I (t), scaling, applying the arccosine function,
applying the known phase offset .phi., and scaling by the pitch,
the proof mass displacement x(t) can be determined by the arccosine
algorithm using equation 103. Similarly, the arcsine algorithm can
be implemented according to equation 104 by measuring C.sub.Q(t),
scaling, applying the arcsine function, adjusting by the phase
offset .phi., and scaling by the pitch to determine x(t).
[0216] The arctangent algorithm can be used as well to determine
displacement. The arctangent algorithm can be implemented according
to equation 105, by measuring the ratio of the capacitances C.sub.Q
and C.sub.I, applying the arctangent function, scaling by the phase
offset .phi., and scaling by the pitch P to determine x(t). The
arcsine algorithm and the arccosine algorithm can be implemented
using only one of the signals C.sub.I and C.sub.Q. In contrast, the
arctangent algorithm uses both signals C.sub.I and C.sub.Q.
Equations 103-105 are written assuming that C.sub.I and C.sub.Q are
90.degree. apart. In general, however, the two signals may have an
arbitrary phase difference .phi.. In this general case, the two
signals can be represented as shown in equations 109-112.
C 1 ( t ) .apprxeq. A cos ( 2 .pi. P x ( t ) ) [ 109 ] C 2 ( t )
.apprxeq. A cos ( 2 .pi. P x ( t ) + .PHI. ) = [ 110 ] A ( cos ( 2
.pi. P x ( t ) ) cos ( .PHI. ) - sin ( 2 .pi. P x ( t ) ) sin (
.PHI. ) ) [ 111 ] C 2 ( t ) C 1 ( t ) = cos ( .PHI. ) - tan ( 2
.pi. P x ( t ) ) sin ( .PHI. ) [ 112 ] ##EQU00054##
[0217] The phase offset .phi. is arbitrary but fixed, so that the
terms cos(.phi.) and sin(.phi.) are fixed constants, and the
inverse tangent of the remaining term, tan(2.pi. x(t)/P), can be
determined using the arctangent algorithm as described above.
[0218] FIG. 26 depicts a block diagram 2600 illustrating signal
flows of the arccosine algorithm. The block diagram 2600 includes a
TDS structure 2602 (e.g., any of TDS structures 105, 207, 506, 604,
606, 1501, 1503, 1606, 1608, 1806, 1808 (FIGS. 1, 2, 5, 6, 15, 16,
and 18)) that generates an analog output signal 2604 (e.g., any of
analog output signals 611, 613, 1626, 1628, 1826, and 1828 (FIGS.
6, 16, and 18)). An AFE 2608 receives the analog output signal
2604. The AFE 2608 can be a charge amplifier (e.g., 618, 1810
(FIGS. 6 and 18)) or a transimpedance amplifier (e.g., 620, 712,
806, 908, 1610 (FIGS. 6, 7, 8, 9, and 16)). An ADC 2612 receives
the output of the AFE 2608 and generates a digital output signal.
The digital output of the ADC 2612 is received by digital circuitry
2622 that performs zero centering, or offsetting. This zero
centering, or offsetting, can include integrating the digital
output of the ADC 2612 over a predetermined time interval to
determine an integral. The integral corresponds to the mean value
over the predetermined time interval. In some examples, the
predetermined time interval can be a period of oscillation of a
proof mass (e.g., 102, 203, 608, 1604, 1804 (FIGS. 1, 2, 6, 16, and
18)). The zero centering can then include subtracting the integral
from the digital output of the ADC 2612.
[0219] The output of the digital circuitry 2622 is received by
digital circuitry 2624 that scales the received signal, which can
include scaling by the quantity A of equations 103-105. Together,
the digital circuitry 2622 and 2624 condition the digital output of
the ADC 2612. The conditioned digital signal generated by digital
circuitry 2624 is received by digital circuitry 2626 that
implements an arccosine function to trigonometrically invert the
conditioned digital signal. Implementing the arccosine function can
include using a lookup table to determine a table entry
corresponding to the conditioned digital signal.
[0220] The output of the arccosine digital circuitry 2626 is
received by phase unwrap digital circuitry 2628. The phase unwrap
circuitry 2628 determines if a phase jump has occurred and adjusts
the digital signal appropriately. Further details of the phase
unwrap circuitry are depicted in FIG. 31. The phase unwrap
circuitry can adjust the digital signal by the phase offset .phi.
as shown in equations 103-105. The output of the phase unwrap
circuitry 2628 is received by scaling circuitry 2630. The scaling
circuitry 2630 scales the digital signal such that it corresponds
to acceleration in units of g. The scaled output of the scaling
circuitry 2630 is received by signal conditioning circuitry 2632
that performs low-pass filtering and resampling to generate output
inertial data 2634. The signal conditioning circuitry 2632 can also
perform multiplication by a geometric dimension. The geometric
dimension can be a pitch of the TDS structure 2602. The output
inertial data 2634 corresponds to an acceleration of the inertial
device (e.g., 100, 202, 602, 1602, 1802 (FIGS. 1, 2, 6, 16, and
18)).
[0221] In some examples, the arccosine block 2626 is replaced with
an arcsine block that implements an arcsine function to
trigonometrically invert the conditioned digital signal.
Implementing the arcsine function can include using a lookup table
to determine a table entry corresponding to the conditioned digital
signal. In some examples, operations are performed in orders
different than depicted in FIG. 26. For example, the scaling 2624
can be performed before the zero centering 2622. In some examples,
the scaling 2624 and zero centering 2622 can be performed on the
digital signals 2616 and 2618 before dividing 2620.
[0222] The arccosine algorithm is described by equations
113-115.
V 1 ( t ) = C 1 cos ( .omega. P ( A cos ( .omega. t ) + x ( t ) ) )
[ 113 ] V n ( t ) = cos ( .omega. P ( A cos ( .omega. t ) + x ( t )
) ) [ 114 ] A cos ( .omega. t ) + x ( t ) = a cos ( V n ( t ) )
.omega. P [ 115 ] ##EQU00055##
[0223] The arcsine algorithm is described by equations 116-118.
V 2 ( t ) = C 2 sin ( .omega. P ( A sin ( .omega. t ) + x ( t ) ) )
[ 116 ] V n ( t ) = sin ( .omega. P ( A sin ( .omega. t ) + x ( t )
) ) [ 117 ] A sin ( .omega. t ) + x ( t ) = a sin ( V n ( t ) )
.omega. P [ 118 ] ##EQU00056##
[0224] FIG. 27 depicts a block diagram 2700 illustrating the signal
flows of the arctangent algorithm. The block diagram 2700 includes
a TDS structure block 2702 (e.g., any of TDS structures 105, 207,
506, 604, 606, 1501, 1503, 1606, 1608, 1806, 1808 (FIGS. 1, 2, 5,
6, 15, 16, and 18)) that generates analog output signals 2704 and
2706 (each, e.g., any of analog output signals 611, 613, 1626,
1628, 1826, and 1828 (FIGS. 6, 16, and 18)). AFE's 2708 and 2710
receive the analog output signals 2704 and 2706, respectively. Each
of the AFE's 2708 and 2710 can be a charge amplifier (e.g., 618,
1810 (FIGS. 6 and 18)) or a transimpedance amplifier (e.g., 620,
712, 806, 908, 1610 (FIGS. 6, 7, 8, 9, and 16)). ADC's 2712 and
2714 receive the output of the AFE's 2708 and 2710, respectively,
and generate digital output signals 2716 and 2718, respectively.
The digital output signals 2716 and 2718 are received by digital
circuitry that divides the two signals 2716 and 2718 to determine a
quotient signal. The quotient signal is received by digital
circuitry 2722 that performs zero centering, or offsetting. This
zero centering, or offsetting, can include integrating the digital
output of the ADC 2712 over a predetermined time interval to
determine an integral. The integral corresponds to the mean value
over the predetermined time interval. The zero centering can then
include subtracting the integral from the digital output of the ADC
2712.
[0225] The output of the digital circuitry 2722 is received by
digital circuitry 2724 that scales the centered signal, which can
include scaling by the quantity A of equations 103-105. Together,
the digital circuitry 2722 and 2724 condition the digital output of
the ADC 2712. The conditioned digital signal generated by circuitry
2724 is received by digital circuitry 2726 that implements an
arctangent function to trigonometrically invert the conditioned
digital signal. Implementing the arctangent function can include
using a lookup table to determine a table entry corresponding to
the conditioned digital signal.
[0226] The output of the arctangent digital circuitry 2726 is
received by phase unwrap digital circuitry 2728. The phase unwrap
circuitry 2728 determines if a phase jump has occurred and adjusts
the digital signal appropriately. Further details of the phase
unwrap circuitry are depicted in FIG. 31. The phase unwrap
circuitry can adjust the digital signal by the phase offset .phi.
as shown in equations 103-105. The output of the phase unwrap
circuitry 2728 is received by scaling circuitry 2730. The scaling
circuitry 2730 scales the digital signal such that it corresponds
to acceleration in units of g. The scaled output of the scaling
circuitry 2730 is received by signal conditioning circuitry 2732
that performs low-pass filtering and resampling to generate output
inertial data 2734. The signal conditioning circuitry 2732 can also
perform multiplication by a geometric dimension. The geometric
dimension can be a pitch of the TDS structure 2702. The output
inertial data 2734 corresponds to an acceleration of the inertial
device (e.g., 100, 202, 602, 1602, 1802 (FIGS. 1, 2, 6, 16, and
18)).
[0227] In some examples, operations are performed in orders
different than depicted in FIG. 27. For example, the scaling 2724
can be performed before the zero centering 2722. In some examples,
the scaling 2724 and zero centering 2722 can be performed on the
digital signals 2716 and 2718 before dividing 2720.
[0228] The arctangent algorithm is described by equations
119-123.
V 1 ( t ) = C 1 cos ( .omega. P ( A sin ( .omega. t ) + x ( t ) ) )
[ 119 ] V 2 ( t ) = C 2 sin ( .omega. P ( A sin ( .omega. t ) + x (
t ) ) ) [ 120 ] V n ( t ) = sin ( .omega. P ( A sin ( .omega. t ) +
x ( t ) ) ) cos ( .omega. P ( A sin ( .omega. t ) + x ( t ) ) ) = [
121 ] tan ( .omega. P ( A sin ( .omega. t ) + x ( t ) ) ) [ 122 ] A
sin ( .omega. t ) + x ( t ) = a tan ( V n ( t ) ) .omega. P [ 123 ]
##EQU00057##
[0229] The arcsine, arccosine, and arctangent algorithms are
similar in that in each, an analog output of a TDS structure is
digitized by an ADC after amplification by one or more AFE's. In
each method, the analog electronics can include any implementation
that converts the physical motion of the sensor to an electronic
signal such as current or voltage. This can include, for example, a
TIA or a CA. The digital output of the ADC is then processed by
digital circuitry to extract the inertial information of interest.
In the arcsine and arccosine algorithms, only one analog signal is
amplified and digitized, reducing electronics, size and power
consumption, in part because only one AFE is required. Also, the
arccosine algorithm only requires one periodic capacitive structure
(or other periodic sense structure), or two if force-balancing of
the proof mass (e.g., 102, 203, 608, 1604, 1804 (FIGS. 1, 2, 6, 16,
and 18)) is desired. In contrast, the arctangent algorithm
amplifies and digitizes two analog signals and requires two AFE's.
The arctangent algorithm requires at least two arrays of TDS
structures separated in spatial phase by 90.degree., or four arrays
if force-balancing of the proof mass (e.g., 102, 203, 608, 1604,
1804 (FIGS. 1, 2, 6, 16, and 18)) is desired. Thus, inertial
devices (e.g., 100, 202, 602, 1602, 1802 (FIGS. 1, 2, 6, 16, and
18)) utilizing the arcsine and arccosine algorithms, instead of the
arctangent algorithm, have reduced complexity and number of
electrical contacts to the TDS structure (e.g., 105, 207, 506, 604,
606, 1501, 1503, 1606, 1608, 1806, 1808 (FIGS. 1, 2, 5, 6, 15, 16,
and 18)). However, the arctangent algorithm has advantages in
determining the phase unwrap, as will be described in more detail
below.
[0230] Once digitized, the signal is scaled and zero-centered, and
an inverse trigonometric function is applied to the data. In the
arctangent algorithm, the two digital signals are divided by each
other, and the quotient is trigonometrically inverted using an
arctangent function. In the arcsine and arccosine algorithms, the
single digitized signal is trigonometrically inverted using an
arcsine or an arccosine function, respectively. In each of the
three methods, the output of the trigonometric inverse function is
phase. However, because the analog input signal is periodic, the
inverse trigonometric functions are not single-valued.
[0231] Because the inverse trigonometric functions are not
single-valued, they can have multiple output values for a given
input value. When inverse trigonometric functions are implemented
in hardware and software, the outputs of these trigonometric
functions are restricted to a single-valued range at the origin.
However, this can result in degeneracy, because the true phase of
the input analog signal may be outside this restricted range. To
arrive at a result that is outside this restricted range for an
inverse trigonometric function, additional processing must be
performed. This additional processing is referred to herein as
phase unwrapping or unwrapping. Unwrapping recreates the original
phase of the input analog signal, which corresponds to the motion
of the proof mass (e.g., 102, 203, 608, 1604, 1804 (FIGS. 1, 2, 6,
16, and 18)). This concept can also apply to non-inertial sensors,
whereby any oscillatory wave form is modified by an input signal to
be detected.
[0232] The disadvantage of the arcsine and arccosine algorithms is
that the phase is more difficult to unwrap compared to the
arctangent algorithm. In the arctangent algorithm, there is a clear
jump in phase near the unwrap boundaries, facilitating detection of
phase wrap. In the arcsine and arccosine algorithms, the phase
simply changes direction at the unwrapped boundaries, requiring a
more involved algorithm to detect a phase wrap event.
[0233] While the arctangent algorithm requires a simpler phase
unwrap algorithm, it requires more analog circuitry than the
arcsine and arccosine algorithms. The arctangent algorithm requires
twice the number of AFE blocks compared to the arcsine and
arccosine algorithms. The arctangent algorithm also requires a
synchronization between ADC's or simultaneous sampling, and two
banks of TDS structures at respective phases of 0.degree. and
90.degree.. If force balancing is also desired, four banks of TDS
structures at 0.degree., 90.degree., 180.degree., and 270.degree.
are required. The arcsine and arccosine algorithms each require
only one AFE, one ADC, and one bank of TDS structures. To perform
force balancing, only two banks of such structures at 0.degree. and
180.degree. are required for the arcsine and arccosine
algorithms.
[0234] FIG. 28 depicts a graph 2800 that shows the digital output
of the arctangent algorithm for a low amplitude of proof mass
oscillation, an amplitude that does not result in phase wrap
events. The graph 2800 includes a digital output curve 2802 that
represents the output of the arctangent block (e.g., 2726 (FIG.
27)) before phase unwrapping. The curve 2802 is in the shape of a
sine wave, representing the oscillation of the proof mass (e.g.,
102, 203, 608, 1604, 1804 (FIGS. 1, 2, 6, 16, and 18)), plus any
offsets induced by inertial forces (or any signal of interest). The
proof mass (e.g., 102, 203, 608, 1604, 1804 (FIGS. 1, 2, 6, 16, and
18)) has been oscillated with an amplitude of 0.4 microns, less
than one-half of the pitch distance of the TDS structure (e.g.,
105, 207, 506, 604, 606, 1501, 1503, 1606, 1608, 1806, 1808 (FIGS.
1, 2, 5, 6, 15, 16, and 18)). Because the oscillation amplitude is
less than one-half the pitch distance (corresponding to a phase of
.pi./2 radians), the digital output curve 2802 is continuous and
without phase wraps.
[0235] FIG. 29 depicts a graph 2900 showing the output of the
arctangent algorithm when the proof mass (e.g., 102, 203, 608,
1604, 1804 (FIGS. 1, 2, 6, 16, and 18)) has an oscillation
amplitude larger than one-half the pitch distance of the TDS
structure (e.g., 105, 207, 506, 604, 606, 1501, 1503, 1606, 1608,
1806, 1808 (FIGS. 1, 2, 5, 6, 15, 16, and 18)). The graph 2900
includes a digital output curve 2902 generated by digital circuitry
implementing the arctangent algorithm (e.g., 2726). The digital
output curve 2902 includes a phase wrap between points 2904 and
2906 and a second phase wrap between points 2908 and 2910. These
phase wraps occur when the proof mass position reaches
displacements that are integer multiples of one-half the pitch of
the TDS structure (e.g., 105, 207, 506, 604, 606, 1501, 1503, 1606,
1608, 1806, 1808 (FIGS. 1, 2, 5, 6, 15, 16, and 18)). The phase
determined from the arctangent algorithm is .pi. or -.pi. radians
at these points, and at the phase wrap, transitions in the opposite
direction by 2.pi. radians. For example, at point 2904, the phase
is .pi., and the output curve 2902 transitions by -2.pi. radians to
a value of -.pi. at point 2906. This transition happens sharply,
and occurs between adjacent data points of the digital output curve
2902. This sharp transition can be easily detected by setting a
threshold to detect large changes in data between adjacent data
points. This sharp transition occurs because the arctangent
function is not single-valued and returns only values between -.pi.
and .pi. for any given input. The phase unwrap block can be used to
detect these phase wraps and remove their effects.
[0236] FIG. 30 depicts a graph 3000 showing a digital output signal
of the arcsine algorithm, where the proof mass has an oscillation
amplitude greater than one-half the pitch, causing phase wraps. The
graph 3000 includes a digital output curve 3002 that is generated
by digital circuitry implementing an arcsine function (e.g., 2626
(FIG. 26)). The digital output curve 3002 has phase wraps at points
3006 and 3008, but these phase wraps are continuous, simply
involving a reversal in direction of the digital output curve 3002.
Digital outputs of the arccosine algorithm exhibit similar phase
wraps as the arcsine algorithm. In the arcsine algorithm, the
absolute value of the digital output signal is used, so the phase
wraps occur when the absolute value of the digital output signal
reaches 0 or .pi./2 radians. In the arccosine algorithm, the phase
wraps occur when the digital output signal reaches 0 or .pi.
radians, regardless of whether the absolute value of the digital
signal is used. However, in the arcsine and arccosine algorithms,
the phase does not jump by 2.pi., but simply changes direction,
with the same rate of change as before the phase wrap. Comparing
the digital output curve 3002 to a simple data jump threshold is
insufficient to detect the phase wrap for the arcsine and arccosine
algorithms, because the digital output curve 3002 remains
continuous across the phase wraps. However, there is still a sharp
transition in the otherwise smoothly changing data.
[0237] This sharp transition can be detected by the following
method. First, the method determines if a sharp transition has
occurred when the digital output curve 3002 has values near 0 or
.pi./2 for the arcsine algorithm, or 0 and .pi. for the arccosine
algorithm. Second, the digital circuitry determines whether the
transition was due to noise or a genuine phase wrap event. Third,
the digital circuitry keeps track of prior phase directions to
maintain continuity of the unwrapped function.
[0238] In some examples, the digital circuitry can determine when a
phase wrap has occurred by monitoring a running difference between
consecutive data points. Over a given time I, if the difference
between the data point at time i and the data point at time i-1 or
the difference between the data point at time i and the data point
of time i+1 is above .pi./2 or below zero, then the digital
circuitry determines that a phase wrap event has occurred in the
arcsine algorithm. If the arccosine algorithm is implemented, the
digital circuitry compares the two differences to zero and .pi..
The digital circuitry then determines between which data points the
phase wrap occurred by comparing neighboring differences. The
digital circuitry can determine that the phase wrap occurred
between the two data points with the smallest difference. The
corresponding succeeding data point is then modified to take into
account the portion of the difference that occurred before and
after the phase wrap. The sign of the slope of the phase is tracked
by altering the sign of the register. The sign is then applied to
either subtract or add subsequent differences in order to
reconstruct the original phase.
[0239] FIG. 31 depicts a method 3100 illustrating phase unwrapping
in the arccosine and arcsine algorithms. At 3104, digital circuitry
receives input data 3102 and determines the time derivative, or
slope. The input data 3102 can comprise a result of trigonometric
inversion. In some examples, determining the time derivative can
include comparing the current value of the signal to the previous
value of the signal and dividing by the difference in time between
the two data points. At 3106, the digital circuitry determines if
the derivative has changed sign. In some examples, the digital
circuitry can do this by comparing the sign of the time derivative
at the current time increment and comparing it to the sign of the
time derivative at the previous time increment. If the derivative
has not changed in sign, at 3108, the digital circuitry stores the
output value at the current time increment as the sign multiplied
by the output value at the previous time increment, added to the
derivative at the current time increment. Because the sign is
reversed each time a phase wrap is detected, as described below,
the step 3108 adjusts the present value and future values of the
output value according to the number of detected phase wrap
events.
[0240] If, at 3106, the digital circuitry determines that the
derivative has changed in sign, the method 3100 proceeds to step
3110. At 3110, the digital circuitry determines whether the sum of
the current derivative and the output value at the last time
increment is greater than .pi.. If the circuitry determines at 3110
that the sum is not greater than .pi., the method 3100 proceeds to
step 3114. At 3114, the digital circuitry determines whether the
sum of the output value at the previous time increment and the
current derivative is less than zero. If, at 3114, the digital
circuitry determines that the sum is not less than zero, no phase
wrap has occurred and the method 3114 proceeds to step 3108. If, at
step 3110, the digital circuitry determines that the sum is greater
than .pi., or, if, at step 3114, the digital circuitry determines
that the sum is less than zero, the method proceeds to steps 3112
and 3116. At 3116, the sign is reversed, such that the new value of
the sign is the opposite of the previously stored value. At 3112,
the digital circuitry determines if the absolute value of the
current derivative is greater than the absolute value of the
previous derivative. If yes, the method 3100 proceeds to steps 3118
and 3120. At 3118, the previous output value is stored as the sign
multiplied by the output value at time i-2 and added to the
derivative at the previous time i-1. At 3120, the digital circuitry
stores the previous derivative as the value obtained from
subtracting the output values at times i-1 and i-2 from 2.pi..
[0241] If, at 3112, the digital circuitry determines that the
absolute value of the derivative at time i is not greater than the
derivative at time i minus 1, the method 3100 proceeds to steps
3118 and 3122. At 3122, the digital circuitry stores the derivative
at time i as a value obtained by subtracting the output value at
time i and the output value at time i-1 from 2.pi.. In this way,
the digital circuitry can implement the method 3100 to unwrap,
rephase and reconstruct the digital output signal without phase
wrap artifacts.
[0242] In some examples, noise in the digital input data 3102 can
be sufficiently high to cause errors in tracking the phase. This
may occur when noise causes the digital signal to temporarily cross
the phase boundaries at zero and/or .pi./2. This may occur in
particular when the noise is much higher than the quantization
level (or bit resolution) of the ADC such that the noise is greater
than the difference between successive data points near the
boundary.
[0243] FIG. 32 depicts an example of phase unwrap error due to
excessive noise at the phase unwrap boundary. FIG. 32 depicts an
acceleration curve 3202 that has been determined using phase
unwrapping, but exhibits a phase unwrap error at point 3204, where
the phase is not correctly tracked due to noise. To overcome this
error, digital circuitry may determine phase crossings by using a
larger data difference threshold. For example, instead of comparing
local data differences to the phase boundary, a larger threshold
may be used. The threshold can be large enough such that no local
noise in the data is large enough to cause a false phase
transition. The magnitude in this threshold is limited by the
full-scale range of the sensor data. In other words, the threshold
must not be so large that an actual signal of interest (such as
acceleration) can cause a false phase transition by falling within
the threshold range. In practice, this can be designed into the
system such that, for a given resonant frequency, the signal of
interest causes the displacement of the oscillator to stay well
within a single pitch distance of the periodic physical structure.
An example of scaled capacitance signals generated by such a design
is shown in FIG. 33.
[0244] Even with proper thresholding, noise can cause occasional
errors with the arcsine and arccosine algorithms. This issue is
specific to the arcsine and arccosine algorithms, in contrast to
the arctangent algorithm. These errors may arise in the output
signal near the phase crossing boundaries, because noise tends to
become magnified near these phase crossing boundaries. These errors
do not occur when using the arctangent algorithm because its sharp
a phase transition make false phase transitions unlikely to occur.
In particular, the error can be highest when a phase boundary is
crossed. This type of error tends to manifest as significant errors
confined to the phase crossing boundary region. Because of this,
these errors can be systematically reduced by interpolating between
neighboring output data points.
[0245] FIG. 33 depicts capacitive signals of an inertial device
(e.g., 100, 202, 602, 1602, 1802 (FIGS. 1, 2, 6, 16, and 18)) with
a proof mass (e.g., 102, 203, 608, 1604, 1804 (FIGS. 1, 2, 6, 16,
and 18)) that is driven to amplitudes that do not cause false phase
transitions. FIG. 33 depicts a capacitive curve 3302 corresponding
to a 0 g acceleration, and a capacitive curve 3304 corresponding to
a 16 g acceleration, which is a full-scale acceleration of the
inertial device. The proof mass is driven at an oscillation
amplitude such that the signal change under acceleration never
exceeds +1 or is less than -1, which would result in a phase
transition after applying the arccosine function. The threshold may
be set that under full acceleration, the phase does not enter the
threshold range near the phase boundary. In this case, the
threshold may be set much greater than the noise level and may not
cause issues with phase tracking.
[0246] When implementing the arcsine, arccosine, and arctangent
algorithms, the proof mass (e.g., 102, 203, 608, 1604, 1804 (FIGS.
1, 2, 6, 16, and 18)) may be driven at nearly arbitrary amplitudes
without having an effect on signal resolution. These methods only
require the proof mass to traverse at least the distance of
one-half the pitch of the TDS structure (e.g., 105, 207, 506, 604,
606, 1501, 1503, 1606, 1608, 1806, 1808 (FIGS. 1, 2, 5, 6, 15, 16,
and 18)). This minimum distance traversal ensures that there is at
least one positive and one negative periodic phase boundary
crossing. This is equivalent to a requirement that the digitized
signal, before applying an inverse trigonometric function and phase
unwrap, be scaled to maximum and minimum amplitudes of +1 and -1,
respectively, as shown in FIG. 33. The proof mass may be driven at
higher amplitudes without any effect on the resolution of the final
output signal. This allows flexibility when designing the system
for a desired full-scale range of the output signal. In some
examples, it is advantageous to drive the proof mass at a minimum
amplitude necessary to achieve a given full-scale range, in order
to minimize the drive voltage and power, as well as to simplify the
phase unwrap algorithm. In addition, drive amplitude does not
affect the noise floor in many cases.
[0247] FIG. 34 depicts capacitance curves of an inertial device
(e.g., 100, 202, 602, 1602, 1802 (FIGS. 1, 2, 6, 16, and 18)) with
a proof mass (e.g., 102, 203, 608, 1604, 1804 (FIGS. 1, 2, 6, 16,
and 18)) driven at two different amplitudes. FIG. 34 depicts a
capacitive curve 3402 of an oscillating proof mass driven to an
amplitude of 0.5 microns and a capacitive curve 3404 of an
oscillating proof mass driven to an amplitude of 3.5 microns. When
the oscillator is driven to an amplitude of 0.5 microns, the proof
mass does not reach a negative phase boundary, and so the
capacitive signal cannot be scaled to maximum and minimum values of
+1 and -1. When the proof mass is driven to an oscillation
amplitude of 3.5 microns, there are multiple phase boundary
crossings, allowing the capacitive signal to be properly scaled to
maximum and minimum values of +1 and -1, respectively.
Additionally, the drive amplitude is such that the digitized
capacitive signal (with no input acceleration) ranges halfway
between phase boundaries, resulting in an optimized full-scale
range. In general, driving the proof mass in quarter-pitch
amplitude increments (that are greater than one-half pitch)
optimizes a given full scale range. The full scale range itself can
be optimized by choosing the resonant frequency of the sensor and
thus its mechanical sensitivity (displacement caused by a given
input acceleration).
[0248] Although it is preferable to drive the proof mass (e.g.,
102, 203, 608, 1604, 1804 (FIGS. 1, 2, 6, 16, and 18)) at
amplitudes greater than one-half pitch of the TDS structure (e.g.,
105, 207, 506, 604, 606, 1501, 1503, 1606, 1608, 1806, 1808 (FIGS.
1, 2, 5, 6, 15, 16, and 18)), this is not necessary. For example,
the arcsine or arccosine algorithms with phase unwrapping can be
used to determine inertial parameters from the scaled capacitive
signal 3402. However, scaling the signal after digitization by the
ADC is more complicated when the oscillator is driven at amplitudes
less than one-half the pitch. One potential implementation for
scaling is a one-time calibration. Another implementation involves
occasionally driving the proof mass to higher amplitudes when the
inertial device (e.g., 100, 202, 602, 1602, 1802 (FIGS. 1, 2, 6,
16, and 18)) is known to be at rest, or during start up of the
inertial device, to measure the appropriate scaling factor.
Conversely, the proof mass oscillation amplitude may be arbitrarily
high. This would increase the number of phase crossings, which must
be tracked. In addition, it is possible to use a full-scale signal
range that causes a displacement that exceeds a pitch or a
half-pitch interval. Accommodating this range requires the digital
circuitry to track phase crossings due to the signal as well as
from the mechanical oscillation.
[0249] FIG. 35 illustrates the error reduction from interpolation.
FIG. 35 depicts an phase error curve 3502 showing phase error of an
inertial device (e.g., 100, 202, 602, 1602, 1802 (FIGS. 1, 2, 6,
16, and 18)) when interpolation is not used, and a phase error
occurred 3504 showing phase error of the same inertial device when
interpolation has been used. As depicted in FIG. 35, interpolation
significantly reduces phase error, and in particular by reducing
the large spikes at single data points.
[0250] FIG. 36 depicts an enlarged view of the phase error curves
3502 and 3504 (FIG. 35). As depicted in FIG. 36, interpolation
removes disproportionately large phase errors at single data
points. With interpolation, some error remains, but this error is
periodic with the motion of the proof mass (e.g., 102, 203, 608,
1604, 1804 (FIGS. 1, 2, 6, 16, and 18)). This error occurs as
spectral artifacts at harmonics of the oscillation frequency of the
proof mass. No such artifacts appear in FIG. 36 below the
oscillation frequency, or in the range of interest for desired
signals (below resonance). However, for high levels of noise and
without interpolation of points of phase boundary crossings,
artifacts may occur in the range of interest.
[0251] In addition to interpolation, error may be reduced further
by implementing a digital low-pass filter before applying the
unwrap algorithm. The sample rate of the ADC may be much greater
than the frequency range of the desired signal. Therefore, most of
the noise is at high frequencies and may be filtered out, provided
the frequency content of the proof mass oscillation is preserved.
In some examples, the low-pass filter can remove noise at
frequencies more than twenty times the drive frequency of the proof
mass. This filtering improves the fidelity of the phase unwrap
algorithm and reduces the overall noise floor.
[0252] With proper thresholding, interpolation, and digital
pre-filtering, the arcsine and arccosine algorithms can have
equivalent noise performance as the arctangent algorithm.
[0253] The arctangent algorithm operates on the output of an ADC to
determine inertial parameters. The arctangent algorithm unfolds
periodic non-linear signal output from the TDS structures (e.g.,
105, 207, 506, 604, 606, 1501, 1503, 1606, 1608, 1806, 1808 (FIGS.
1, 2, 5, 6, 15, 16, and 18)), recovering a digitized representation
of the motion of the proof mass (e.g., 102, 203, 608, 1604, 1804
(FIGS. 1, 2, 6, 16, and 18)). The low-frequency displacements that
are induced by inertial forces are the desired signals and can be
isolated from the oscillation of the proof mass (and any other
higher frequency motion) by digital low-pass filtering.
[0254] In some examples, the arctangent algorithm requires in-phase
(I) and quadrature (Q) signals to be generated by the inertial
device (e.g., 100, 202, 602, 1602, 1802 (FIGS. 1, 2, 6, 16, and
18)). These signals have intrinsic phase separation of 90.degree..
These I and Q signals can be produced by arrays of periodic
structures that are offset by 90.degree. or one-fourth the pitch of
the periodic structures. In some examples, the phase separation
need not be exactly 90.degree., in which case the modified
equations can be used. In some examples, to prevent imbalances from
capacitive forces and to employ a differential AFE amplifier to
reject common-mode noise, four arrays of TDS structures with
0.degree., 180.degree., 90.degree. and 270.degree. phase offsets
may be used as shown in FIG. 37.
[0255] FIG. 37 depicts an inertial device 3700 with TDS structures
(e.g., 105, 207, 506, 604, 606, 1501, 1503, 1606, 1608, 1806, 1808
(FIGS. 1, 2, 5, 6, 15, 16, and 18)) that have four different phase
offsets. The inertial device 3700 includes a proof mass 3702 that
is oscillated along the x axis by drive combs 3704a, 3704b, 3704c,
and 3704d (collectively, drive combs 3704). FIG. 37 depicts a
coordinate system 3703 with an x axis, a y axis perpendicular to
the x axis, and a z axis perpendicular to each of the x and y axes.
The proof mass 3702 is connected to springs and anchors 3706a and
3706b (collectively, anchors 3706). The inertial device 3700 also
includes anchors 3708a, 3708b, 3708c, and 3708d (collectively,
anchors 3708). The anchors 3706 and 3708 are connected to a top
layer and/or bottom layer (not shown).
[0256] The inertial device 3700 includes 0.degree. TDS structures
3710a and 3710b (collectively, TDS structures 3710), 90.degree. TDS
structures 3712a and 3712b (collectively, TDS structures 3712),
180.degree. TDS structures 3714a and 3714b (collectively, TDS
structures 3714), and 270.degree. TDS structures 3716a and 3716b
(collectively, TDS structures 3716). FIG. 37 also depicts areas of
interest 3718, 3720, 3722, and 3724.
[0257] FIG. 37 depicts the enlarged views 3726, 3728, 3730, and
3732, which are enlarged views of the areas of interest 3718, 3720,
3722, and 3724, respectively. The view 3726 depicts a portion of
the TDS structure 3710a and shows moveable beams 3734a and 3738a.
The moveable beams 3734a and 3738a move along the x axis with
respect to a fixed beam 3736a. The view 3726 depicts the proof mass
3702 in the neutral position, and the teeth of the fixed beam 3736a
are aligned with the teeth of the moveable beams 3734a and 3738a,
corresponding to a spatial phase of 0.degree. and 0 radians.
[0258] The view 3728 depicts a portion of the TDS structure 3712a
and shows moveable beams 3734b and 3738b. The moveable beams 3734b
and 3738b move along the x axis with respect to a fixed beam 3736b.
The view 3728 depicts the proof mass 3702 in the neutral position,
and the teeth of the fixed beam 3736b are offset from the teeth of
the moveable beams 3734b and 3738b by one-fourth of the pitch
distance of the TDS structure 3712a, corresponding to a spatial
phase of 90.degree. and .pi./2 radians.
[0259] The view 3730 depicts a portion of the TDS structure 3714a
and shows moveable beams 3734c and 3738c. The moveable beams 3734c
and 3738c move along the x axis with respect to a fixed beam 3736c.
The view 3730 depicts the proof mass 3702 in the neutral position,
and the teeth of the fixed beam 3736c are offset from the teeth of
the moveable beams 3734c and 3738c by one-fourth of the pitch
distance of the TDS structure 3714a, corresponding to a spatial
phase of 180.degree. and .pi. radians.
[0260] The view 3732 depicts a portion of the TDS structure 3716a
and shows moveable beams 3734d and 3738d. The moveable beams 3734d
and 3738d move along the x axis with respect to a fixed beam 3736d.
The view 3732 depicts the proof mass 3702 in the neutral position,
and the teeth of the fixed beam 3736d are offset from the teeth of
the moveable beams 3734d and 3738d by one-fourth of the pitch
distance of the TDS structure 3716a corresponding to a spatial
phase of 270.degree. and 3.pi./2 radians.
[0261] The capacitance of the 0.degree. of the TDS structure 3710
as a function of displacement of the proof mass 3702 is shown by
equation 124.
C ( x ) = { A + B cos [ 2 .pi. P x ] + C cos [ 4 .pi. P x ] + D cos
[ 6 .pi. P x ] + E cos [ 8 .pi. P x ] + F cos [ 10 .pi. P x ] + G
cos [ 12 .pi. P x ] } [ 124 ] ##EQU00058##
[0262] Because the variables C, D, E, F, and G are approximately
two orders of magnitudes of the variables A and B, equation 124 can
be approximated by equation 125. While equation 124 more accurately
captures the capacitance behavior of the TDS structure 3710, the
simpler equation 125 will be used for the conceptual analysis
below. The pitch of the teeth of the TDS structure 3710 is
indicated by the variable P. The capacitance behavior of the TDS
structures 3710, 3712, 3714, and 3716 can be modeled using equation
125, with spatial phase offsets in quarter-pitch increments as
shown in equations 126-129.
C ( x ) x A + B cos [ 2 .pi. P x ] [ 125 ] C 0 ( t ) = C [ x ( t )
] [ 126 ] C 90 ( t ) = C [ x ( t ) + P 4 ] [ 127 ] C 180 ( t ) = C
[ x ( t ) + P 2 ] [ 128 ] C 270 ( t ) = C [ x ( t ) + 3 P 4 ] [ 129
] ##EQU00059##
[0263] The capacitance can be expressed as a function of time by
substituting the proof mass (e.g., 102, 203, 608, 1604, 1804 (FIGS.
1, 2, 6, 16, and 18)) motion x(t) into equation 125 as shown in
equation 130.
C ( t ) = C [ x ( t ) ] .apprxeq. A + B cos [ 2 .pi. P x ( t ) ] [
130 ] ##EQU00060##
[0264] Using two matched differential AFE's such as transimpedance
or charge amplifiers, the capacitances of the 0.degree. TDS
structure 3710 and the 180.degree. TDS structure 3714 are combined
to define the in-phase signal (I) as shown in equation 131.
Similarly, the capacitance signals of the 90.degree. TDS structure
3712 and the 270.degree. TDS structure 3716 are combined to define
the quadrature signal (Q) as shown in equation 132.
C 1 ( t ) = C 0 ( t ) - C 180 ( t ) .apprxeq. 2 B cos [ 2 .pi. P x
( t ) ] [ 131 ] C Q ( t ) = C 90 ( t ) - C 270 ( t ) .apprxeq. 2 B
sin [ 2 .pi. P x ( t ) ] [ 132 ] ##EQU00061##
[0265] FIGS. 38, 39, and 40 depict the capacitance signals of
equations 131 and 132 at various drive amplitudes of the proof mass
(e.g., 102, 203, 608, 1604, 1804 (FIGS. 1, 2, 6, 16, and 18)). FIG.
38 depicts an in-phase capacitance curve 3802 and a quadrature
capacitance curve 3804 at a drive amplitude of 2 microns. FIG. 39
depicts an in-phase capacitance curve 3902 and a quadrature
capacitance curve 3904 at a drive amplitude of seven microns. FIG.
40 depicts an in-phase capacitance curve 4002 and a quadrature
capacitance curve 4004 at a drive amplitude of 12 microns. As can
be seen in FIGS. 38-40, when the drive amplitude increases, the
peak capacitance does not change, but the frequency of the
capacitance signal does change. This is true even though the
oscillation frequency of the proof mass has not changed.
[0266] The displacement of the proof mass (e.g., 102, 203, 608,
1604, 1804 (FIGS. 1, 2, 6, 16, and 18)) can be determined by
dividing equations 131 to produce equation 133. Thus, digital
circuitry can perform the operations of equation 133 on received
capacitance signals C.sub.Q and C.sub.I to determine proof mass
displacement.
x ( t ) = P 2 .pi. a tan [ C Q ( t ) C t ( t ) ] [ 133 ]
##EQU00062##
[0267] The digital representation of the motion of the proof mass
(e.g., 102, 203, 608, 1604, 1804 (FIGS. 1, 2, 6, 16, and 18)) can
include different frequency components, including contributions
from the drive comb actuation, from inertial forces, and from
acoustic coupling. These components are illustrated in equation
134.
x(t)=Asin(.omega..sub.0t)+x.sub.Inertial(t)X.sub.Acoustic(t)
[134]
[0268] The first term in equation 134, Asin (.omega..sub.0t),
represents the resonant motion of the proof mass caused by comb
drives. This component can be extracted using a digital band-pass
filter. In some examples, this digital band-pass filter can utilize
a third order Butterworth filter centered at two kilohertz with
cut-offs at 2.25 and 1.75 kHz. These filter parameters can be used
for a drive frequency of 2 kHz. The oscillator amplitude can be
isolated from this filtered digital signal using an envelope
detector. The third term in equation 134 represents high frequency
motion (e.g., 200 Hz-20 kHz), caused by acoustic coupling from a
speaker. These signals are at frequencies above the inertial
signals, but if acoustic signals exist in the band of the band pass
filter, there can corrupt the amplitude signal.
[0269] The second term in equation 134, x.sub.Inertial (t),
represents low-frequency motion of the proof mass (e.g., less than
200 Hz) caused by inertial forces acting on the inertial device.
Motion in this frequency range is the desired measurement. The
inertial component of the signal is isolated using a digital
low-pass filter. In some examples, the low-pass filter can be a
fourth order Butterworth filter with a 200 Hz cutoff. The inertial
acceleration is thus given by equation 135, where
.omega..sub.0.sup.2 represents the square of the natural frequency
of the proof mass. In some examples, .omega..sub.0.sup.2 represents
the square of the drive frequency of the proof mass.
a(t)=.omega..sub.0.sup.2x.sub.Inertial(t) [135]
[0270] The resonant frequency of the proof mass can be measured in
real time because the closed loop drive can accurately track
resonance. Initial calibration can be used to determine the
resonant frequency. The relative change in sensitivity over time
can be tracked by measuring the closed loop drive frequency, with
some initial calibration. The relative change in sensitivity can
include a fixed offset. If the fixed offset drifts with time, this
can affect accuracy of the measurement of inertial parameters. In
the arcsine, arccosine, and arctangent algorithms, the unwrap
accuracy does not depend on knowledge of the actual resonant
frequency. The unwrap accuracy in these algorithms only depends on
an accurate measurement of the drive frequency. However, knowledge
of the actual resonant frequency does affect the scaling of the
unwrapped output to units of `g` in the arcsine, arccosine, and
arctangent algorithms, as well as in the cosine algorithm.
[0271] FIGS. 41 and 42 depict analog output signals of differential
charge amplifiers at proof mass (e.g., 102, 203, 608, 1604, 1804
(FIGS. 1, 2, 6, 16, and 18)) oscillation of 7 microns and 4
microns, respectively. FIG. 41 includes an in-phase analog signal
4102 and an out-of-phase analog signal 4104. FIG. 42 includes an
in-phase analog signal 4202 and an out-of-phase analog signal 4204.
The signals 4102, 4104, 4202, and 4204 represent outputs of one or
more matched analog to digital converters that can use sigma-delta
direct conversion, and/or successive-approximation methods. The
curves 4102, 4104, 4202, and 4204 are sampled at 400 kHz over one
period of a 2 kHz resonant oscillation of the proof mass. In some
examples, the arctangent algorithm requires the amplitudes of the I
and Q signals to be equalized. One way to achieve this is to use a
peak detection algorithm to determine the maximum amplitude of each
signal and then scale each signal appropriately. In addition, the
mean value, or DC component, of both I and Q signals should be
zero. One way to achieve this is to subtract off the integrated
value, or mean value, over one period of oscillation of the proof
mass. Another way to achieve the zero mean is to constrain the
drive amplitude to discrete levels at which the I and Q signals are
naturally zero-valued. In particular, this condition requires that
the quantity 2.pi.* Amplitude/Pitch equal a zero of the zero order
Bessel function of the first kind. After this scaling and
mean-value adjusting, the I and Q signals can be divided as shown
in equation 136.
C Q ( t ) C 1 ( t ) = tan [ 2 .pi. P x ( t ) ] [ 136 ]
##EQU00063##
[0272] FIG. 43 depicts the ratio of Q and I signals given by
equation 136. FIG. 43 includes a curve 4302 that illustrates this
ratio and includes phase wrap events. To recover displacement
information of the proof mass, equation 136 can be inverted,
resulting in equation 137.
x ( t ) P = 1 2 .pi. arctan [ C Q ( t ) C I ( t ) ] [ 137 ]
##EQU00064##
[0273] However, applying the arctangent function of equation 137
requires applying a phase unwrap function. In some examples, this
function monitors the output of the arctangent function and adds
multiples of .+-.2.pi. when absolute jumps between adjacent data
points are greater than or equal to .pi. radians.
[0274] FIG. 44 depicts the arctangent of the Q/I ratio without
unwrapping, as shown by curve 4402.
[0275] FIG. 45 depicts the proof mass position after unwrapping.
FIG. 45 includes a calculated displacement curve 4502 that
represents a digital estimate of the proof mass position. After
unwrapping, the curve 4502 is a smooth sinusoidal curve without
phase wraps.
[0276] FIG. 46 depicts an enlarged view of a portion of FIG. 45,
showing the difference between the true displacement and the
digital estimate. FIG. 46 includes a displacement curve 4602 that
represents the true displacement of the proof mass. FIG. 46 also
includes a digital estimate curve 4604 that represents the digital
estimate of the position of the proof mass. The curve 4604 drawn in
FIG. 46 is an enlarged view of the curve 4502 shown in FIG. 45.
FIG. 46 shows the quantization error inherent in any digital
representation of a continuous variable.
[0277] The arcsine, arccosine, and arctangent algorithms described
herein are useful because they produce a digitized, accurate
representation of oscillator position as a function of time, scaled
by the pitch of a TDS structure (e.g., 105, 207, 506, 604, 606,
1501, 1503, 1606, 1608, 1806, 1808 (FIGS. 1, 2, 5, 6, 15, 16, and
18)). These methods have a bandwidth higher than the drive
frequency of the proof mass (e.g., 102, 203, 608, 1604, 1804 (FIGS.
1, 2, 6, 16, and 18)). This high bandwidth prevents inputs from
frequencies higher than the resonant frequency from coupling or
aliasing into the inertial low-frequency band and affecting the
inertial acceleration measurements. In addition, the arcsine,
arccosine, and arctangent algorithms accurately map the motion of
the proof mass, regardless of whether the motion is sinusoidal.
Thus, the arcsine, arccosine, and arctangent algorithms will
accurately recover the displacement and acceleration signals,
despite spring non-idealities or high-frequency vibrational
coupling.
[0278] The cosine, arcsine, arccosine, and arctangent algorithms
are relatively immune to 1/f noise caused by electronics amplifiers
and filters. The algorithms essentially encode the acceleration
information on a higher frequency signal, thus up-modulating the
acceleration information. Thus, low frequency drift of the
electronics does not impact accuracy or drift of the acceleration
measurement. The algorithms effectively remove offset and gain
drift of electronics from acceleration measurement accuracy. As a
result, only white noise significantly impacts resolution of the
algorithms (but does not impact drift).
[0279] In some examples, analog and/or digital circuitry of the
inertial device (e.g., 100, 202, 602, 1602, 1802 (FIGS. 1, 2, 6,
16, and 18)) is included in a single mixed-signal
application-specific integrated circuit (ASIC) located on a single
substrate. In other examples, analog and/or digital circuitry of
the inertial device (e.g., 100, 202, 602, 1602, 1802 (FIGS. 1, 2,
6, 16, and 18)) is distributed between multiple integrated
circuits. The multiple integrated circuits can all be located on a
single substrate. In some examples, the multiple integrated
circuits can be distributed across multiple substrates that are
electrically connected. In some examples, some or all of the
inertial device (e.g., 100, 202, 602, 1602, 1802 (FIGS. 1, 2, 6,
16, and 18)) is implemented using one or more digital
processors.
[0280] The systems described herein can be fabricated using MEMS
and microelectronics fabrication processes such as lithography,
deposition, and etching. The features of the inertial device (e.g.,
100, 202, 602, 1602, 1802 (FIGS. 1, 2, 6, 16, and 18)) are
patterned with lithography and selected portions are removed
through etching. Such etching can include deep reactive ion etching
(DRIE) and wet etching. In some examples, one or more intermediate
metal, semiconducting, and/or insulating layers are deposited. The
base wafer can be a doped semiconductor such as silicon. In some
examples, ion implantation can be used to increase doping levels in
regions defined by lithography. The proof mass (e.g., 102, 203,
608, 1604, 1804 (FIGS. 1, 2, 6, 16, and 18)) and TDS structures
(e.g., 105, 207, 506, 604, 606, 1501, 1503, 1606, 1608, 1806, 1808
(FIGS. 1, 2, 5, 6, 15, 16, and 18)) can be defined in a substrate
silicon wafer, which is then bonded to top and bottom cap wafers,
also made of silicon. Encasing the proof mass in this manner allows
the volume surrounding the mass to be evacuated. In some examples,
a getter material such as titanium is deposited within the
evacuated volume to maintain a low pressure throughout the lifetime
of the device. This low pressure enhances the quality factor of the
resonator. From the proof mass and TDS structures (e.g., 105, 207,
506, 604, 606, 1501, 1503, 1606, 1608, 1806, 1808 (FIGS. 1, 2, 5,
6, 15, 16, and 18)), conducting traces are deposited using metal
deposition techniques such as sputtering or physical vapor
deposition (PVD). These conducting traces electrically connect
active areas of the proof mass and TDS structures to the
microelectronic circuits depicted in FIG. 1. Similar conducting
traces can be used to electrically connect the microelectronic
circuits depicted in FIG. 1 to each other. The fabricated MEMS and
microelectronic structures can be packaged using semiconductor
packaging techniques including wire bonding and flip-chip
packaging.
[0281] As used herein, the term "memory" includes any type of
integrated circuit or other storage device adapted for storing
digital data including, without limitation, ROM, PROM, EEPROM,
DRAM, SDRAM, DDR/2 SDRAM, EDO/FPMS, RLDRAM, SRAM, flash memory
(e.g., AND/NOR, NAND), memrister memory, and PSRAM.
[0282] As used herein, the term "digital circuitry" is meant
generally to include all types of digital processing devices
including, without limitation, digital signal processors (DSPs),
reduced instruction set computers (RISC), general-purpose (CISC)
processors, microprocessors, field programmable gate arrays
(FPGAs), PLDs, reconfigurable compute fabrics (RCFs), array
processors, secure microprocessors, and application-specific
integrated circuits ASICs. Such digital processors may be contained
on a single unitary integrated circuit die, or distributed across
multiple components.
[0283] From the above description of the system it is manifest that
various techniques may be used for implementing the concepts of the
system without departing from its scope. For example, in some
examples, any of the circuits described herein may be implemented
as a printed circuit. Further, various features of the system may
be implemented as software routines or instructions to be executed
on a processing device (e.g. a general purpose processor, an ASIC,
field programmable gate array (FPGA), etc.) The described
embodiments are to be considered in all respects as illustrative
and not restrictive. It should also be understood that the system
is not limited to the particular examples described herein, but can
be implemented in other examples without departing from the scope
of the claims.
[0284] Similarly, while operations are depicted in the drawings in
a particular order, this should not be understood as requiring that
such operations be performed in the particular order shown or in
sequential order, or that all illustrated operations be performed,
to achieve desirable results.
* * * * *